Galatians 1:13-14 New King James Version (NKJV)
13 For you have heard of my former conduct in Judaism, how I persecuted the church of God beyond measure and tried to destroy it. 14 And I advanced in Judaism beyond many of my contemporaries in my own nation, being more exceedingly zealous for the traditions of my fathers.
1 On Paul’s “former conduct” in “Judaism.” It’s clear to me that he couldn’t have said that he is no longer a jew because Christianity didn’t exist yet. Also, for the following reasons:
Acts 28:17 New King James Version (NKJV)
17 And it came to pass after three days that Paul called the leaders of the Jews together. So when they had come together, he said to them: “Men and brethren, though I have done nothing against our people or the customs of our fathers, yet I was delivered as a prisoner from Jerusalem into the hands of the Romans,Acts 23:6 New King James Version (NKJV)
6 But when Paul perceived that one part were Sadducees and the other Pharisees, he cried out in the council, “Men and brethren, I am a Pharisee, the son of a Pharisee; concerning the hope and resurrection of the dead I am being judged!”Acts 22:3 New King James Version (NKJV)
3 “I am indeed a Jew, born in Tarsus of Cilicia, but brought up in this city at the feet of Gamaliel, taught according to the strictness of our fathers’ law, and was zealous toward God as you all are today.Acts 25:8 New King James Version (NKJV)
8 while he answered for himself, “Neither against the law of the Jews, nor against the temple, nor against Caesar have I offended in anything at all.”
Consider also Paul’s statement that he is a member of what is called a sect of Judaism before the Romans:
13 Nor can they prove the things of which they now accuse me. 14 But this I confess to you, that according to the Way which they call a sect, so I worship the God of my fathers, believing all things which are written in the Law and in the Prophets. (Acts 24 NKJV)
Compare with:
https://hermeneutics.stackexchange.com/questions/7682/does-nazarene-sect-mean-nazarene-cult
. . . Ἰουδαϊσμῷ The rendering of this word in our versions, Jewish religion, is unfortunate: it implies a definite separation between the two religions which did not then exist, for Christians were still habitual worshippers in the synagogue; and it puts this view into the mouth of Paul, who steadfastly persisted in identifying the faith of Christ with the national religion. The word Ἰουδαϊζειν denotes the adoption of Jewish habits, language, or policy (cf. Galatians 2:14). So here Ἰουδαϊσμός denotes Jewish partisanship . . .
(Expositor’s Greek Testament)
. . . the Jews’ religion] One word in the original, which does not occur elsewhere in the N. T. except in Galatians 1:14. From the use of the corresponding verb, we may regard it as referring not to the religion revealed to the Jews in the writings of Moses and the prophets, but that which was its actual development in St Paul’s day, when the word of God had been overlaid and ‘made of none effect’ by the traditions of the Scribes and Pharisees, and the puerile conceits of the Rabbinic expositors. . . .(Cambridge Bible for Schools and Colleges) https://biblehub.com/commentaries/galatians/1-13.htm
“former” modifies “conduct” not “Judaism” (G2454)
So Paul had a former behavior in “Judaism” and now he has a different behavior in “Judaism” after learning the gospel of Christ.
- the word translated “Judaism” is only twice in the entire NT (Gal 1:13-14) and LXX (excluding Maccabees): https://studybible.info/search-interlinear/strongs/2454
13–14 For you have heard of my former manner of life in Judaism, how I used to persecute the church of God beyond measure and tried to destroy it; and I was advancing in Judaism beyond many of my contemporaries among my countrymen, being more extremely zealous for my ancestral traditions.
The word “Judaism” ( jIoudai>smov~, ioudaismos) is used only here (twice, once in v. 13 and once in v. 14) in the whole Apostolic Scriptures. It is found only five times in the Lxx (2Mac 2.21; 8.1; 14.38(2x); 4Mac 4.26) and then only in the Maccabees. Y. Amir, in a study entitled “The Term Ioudaismos: A Study in Jewish-Hellenistic Self-Identification,”50 comes to the conclusion that the word means a “a sort of fenced-off area in which Jewish lives are led.”
https://www.torahresource.com/radio-files/study-in-galatians/galatians_commentary.pdf
the word this is derived from G2450 (verb form) which is only used in Ester 8:17 and Galatians 2:14 https://studybible.info/search-interlinear/strongs/G2450 In Gal 2:14 it is used to denote specific Jewish practices. In Ester 8:17 it is used for “jewish-like” in “were circumcised and were jewish-like” indicating that it might mean something other than circumcision. A GREEK-ENGLISH LEXICON OF THE SEPTUAGINT defines it as:
+ V 0-0-0-1-0=1
Est 8,17
to side with or to imitate the Jews; neol
http://www.glasovipisma.pbf.rs/phocadownload/knjige/greek%20lexicon%20for%20the%20septuagint.pdf
Dunn thinks that the term may indicate something other than “circumcision” (=becoming a proselyte), since its only other use (the Lxx of Esther 8:17) has both the term “circumcised” as well as “made themselves Jews” (the verb ‘to circumcise” is lacking in the Hebrew). But the Lxx phrase (“and many of the Gentiles were circumcised, and became Jews, for fear of the Jews”) may well be simply a commentary on “becoming a Jew.” From the Lxx translators’ standpoint, this surely involved the ritual of the proselyte. It hardly seems possible that Paul would have so sharply denounced Peter if he was simply trying to persuade the Gentiles to take on Jewish customs. This hardly goes
contrary to the gospel. Rather, it seems to me far more likely that Peter, for what ever reasons, was attempting to sway the Gentile believers over to the viewpoint of the “party of the circumcision,” that full covenant membership was only available to Jews.
From the viewpoint of the influencers, the whole matter turned on the observance of established halachah. But for Paul, the issue was that of the gospel: “But when I saw that they were not straightforward about the truth of the gospel ….” The word translated “straightforward” by the NASB (“not acting in line,” NIV; “their conduct was not in step,” ESV) is interesting. It is ojrqopodevw, orthopodeõ, being made of two words; ortho, meaning “straight” (note our English “orthodontist”) and pous, “foot.” The obvious idea is “to walk in a straight path,” “to be on the right road.” Our modern idiom, “walk a straight line” fits the meaning well. It was not that Peter and those he was following were denying the gospel, nor attempting to undermine it directly. Rather, their approach to this whole matter was a detour from the gospel, and one that Paul feared would so sidetrack the Gentile believers as to keep them from reaching the goal
https://www.torahresource.com/radio-files/study-in-galatians/galatians_commentary.pdf
In the context of Galatians 2, G2450 may be related to a halacha or Jewish practice: 1 a false gospel or good news (how you are saved) and 2 justifying yourself by works of the law. . . This seems to fall outside of torah vs. non-torah observant, rather it is a certain take the Torah or on Judaism. (mainly the false idea that you needed to become a Jew to be saved) In this case it was false but that doesn’t mean the word G2450 implies falsity. See below:
13 And the rest of the Jews also played the hypocrite with him, so that even Barnabas was carried away with their hypocrisy.
14 But when I saw that they were not straightforward about the truth of the gospel, I said to Peter before them all, “If you, being a Jew (G2450), live in the manner of Gentiles and not as the Jews, why do you compel Gentiles to live as Jews? 15 We who are Jews by nature, and not sinners of the Gentiles, 16 knowing that a man is not justified by the works of the law but by faith in Jesus Christ, even we have believed in Christ Jesus, that we might be justified by faith in Christ and not by the works of the law; for by the works of the law no flesh shall be justified. (Gal 2)
https://www.torahresource.com/radio-files/study-in-galatians/galatians_commentary.pdf
Note: “we are Jews by birth and not Gentile sinners” is tongue-and-cheek (explained later)
Peter’s hypocrisy consisted of his having engaged in table fellowship
with the Gentiles when unobserved by the Jerusalem folk, but separating from the Gentiles when the group from James arrived, and even compelling them to submit to proselytism in order to be accepted by the party of the circumcision. The Greek has ijoudai÷zw, ioudaizõ, “to live like a Jew,” used only here in the Apostolic Scriptures. The Lxx utilizes this same verb in Esther 8:17 (the only time found in the Lxx) to translate the hapax legoumena . . . , mityahadim, “made themselves Jews.” Though the term is used only these two times in biblical literature, the meaning is clear: Peter had been swayed by the “party of the circumcision” to compel (ajnagkavzw, anagkazõ45) the Gentiles to submit to the ritual of a proselyte. Interestingly, Paul used this same word (Acts 26:11) to describe his attempts to “force” the believers in Yeshua to blaspheme in order to have a sure judgment against them
https://www.torahresource.com/radio-files/study-in-galatians/galatians_commentary.pdf
2 How was Peter compelling Gentiles to live as Jews? By making them undergo the proselyte ritual in order to consider them saved and hence worthy of table fellowship. Similar to how Paul was put in a situation where he was “compelled” to call on Cesar where same word is used: Acts 28:19 https://studybible.info/search-interlinear/strongs/G315
3 The Revelation that Paul had was about Yeshua’s salvation for the nations. It seems like this revelation started on the road to Damascus and continued afterward:
15 But when God thought well to separate me from out of the belly of my mother, and called me by his favor, 16 to reveal his son in me, that I should announce him good news among the nations; immediately I did not confer with flesh and blood; (Gal 1 ABP)
4 A suggestion on how the New Covenant is different:
The New Covenant covenant is different in that it is God writing the law on the heart rather than man. This I think is significant. Let’s say I make an agreement with you that you will build a house with certain specifications and that I will pay you a certain amount of money. Then I propose a different agreement where the only difference is that I will build the house instead of you. That’s pretty significant, even if I don’t change the money or any of the other specifications. I think maybe the other difference is that God is able to write the law on the heart much better than man (hence why no one will need to teach another which is also different): this is possibly why it also differs in that God will write the law “on your inward parts” or “inside of you” in the New Covenant (maybe a deeper writing than just the mind)
Compare inward parts to heart (I thought this was kind of interesting)
There seems to be some significance to the writing surface and instrument used in metaphors involving writing: As a side note  Jer 17:13 is used to explain what Jesus wrote in the dust: https://www.youtube.com/watch?v=rZSG7p5DQ-M
 Jer 17:1 The sin of Judah is written H3789 with a pen of iron, and with the point of a diamond: it is graven upon the table of their heart, and upon the horns of your altars; (KJV)
Jer 17:13 O LORD, the hope of Israel, all that forsake thee shall be ashamed, and they that depart from me shall be written H3789 in the earth, because they have forsaken the LORD, the fountain of living waters. (KJV)
Compare with:
Deu 30:14 But the word is very nigh unto thee, in thy mouth, and in thy heart, that thou mayest do it. (KJV)
It says in proverbs to either write God’s law on your heart or to write the instructions of whoever is writing proverbs on your heart:
 Pro 3:3 Let not mercy and truth forsake thee: bind them about thy neck; write H3789 them upon the table of thine heart: (KJV)
 Pro 7:3 Bind them upon thy fingers, write H3789 them upon the table of thine heart. (KJV)
Compare with:
Jer 31:33 But this shall be the covenant that I will make with the house of Israel; After those days, saith the LORD, I will put my law in their inward parts, and write H3789 it in their hearts; and will be their God, and they shall be my people. (KJV)
It seems like God would have better instruments than us to write the law. Keil and Deilitzche in their commentary on Deut 10:6 make the observation that the writing surface will be entirely different as well:
The Lord will then circumcise their heart, and the heart of their children (see Deuteronomy 10:16), so that they will love Him with all their heart. When Israel should turn with true humility to the Lord, He would be found of them, – would lead them to true repentance, and sanctify them through the power of His grace, – would take away the stony heart out of their flesh, and give them a heart of flesh, a new heart and a new spirit, – so that they should truly know Him and keep His commandments (vid., Ezekiel 11:19; Ezekiel 36:26; Jeremiah 31:33. and Deuteronomy 32:39.). “Because of thy life,” i.e., that thou mayest live, sc., attain to true life. The fulfilment of this promise does not take place all at once. It commenced with small beginnings at the deliverance from the Babylonian exile, and in a still higher degree at the appearance of Christ in the case of all the Israelites who received Him as their Saviour. Since then it has been carried on through all ages in the conversion of individual children of Abraham to Christ; and it will be realized in the future in a still more glorious manner in the nation at large (Romans 11:25.). The words of Moses do not relate to any particular age, but comprehend all times. For Israel has never been hardened and rejected in all its members, although the mass of the nation lives under the curse even to the present day.
https://biblehub.com/commentaries/kad/deuteronomy/30.htm
5 Something interesting “We who are Jews by nature, and not sinners of the Gentiles” might be Paul being sarcastic. Hence, he is also speaking against the idea of “works of the law” in this case going through the ritual of proselyte and becoming a “Jew by nature” is not going to automatically save you are make you not a sinner.
Paul quotes (perhaps a bit “tongue-in-cheek”) the primary premise of the party of the circumcision: “we are Jews by birth and not Gentile sinners.” This is not Paul’s perspective, but that of the influencers. Dunn agrees:
This language rings oddly on the lips of Paul, until we realize what hewas doing. Paul was putting himself in the shoes of a typical Jew who looked out at the rest of the world as outside the realm of God’s covenant righteousness and sinful (cf. Eph 2:12). More to the point, he was using the language of typical Jewish factionalism, which was ready to condemn those Jews who disagreed with the sect’s interpretation of what the law required as ‘sinners’— outside their sectarian understanding of the covenant, which meant, of course, from the sectarian viewpoint, outside the covenant. In fact, Paul was probably echoing the language used by the ‘individuals from James’ when they spoke against the Jewish Christians’ table-fellowship with the Gentile believers: such table-fellowship with ‘Gentile sinners’ was unacceptable.46
Thus, when Paul writes, “we are Jews by birth and not Gentile sinners,” he is deliberately using the language of those who were distancing themselves from the Gentiles, encouraging them to become proselytes in order to leave the status of “sinner” and enter the circle of “Jews by birth.”
https://www.torahresource.com/radio-files/study-in-galatians/galatians_commentary.pdf
Conclusion: I think G2454 and G2450 refers to practice or halacha of Judaism. That is, Paul had abandoned his former practice in Judaism that viewed converting to Judaism (or jewish-like) as a prerequisite for salvation. To quote Tim Hegg:
There is no doubt that Paul made a clear distinction between his former life, lived under the acceptance of the prevailing Pharisaic belief that Jewish status rendered one a member of the covenant, and his current life lived in the reality of the risen Messiah. But such a distinction said nothing about the place of the divinely inspired Torah, and its central importance in the life of the believer. What it did contrast, however, was the life of faith in Messiah Yeshua and the
message of the influencers which insisted upon Jewish status as a prerequisite for covenant membership.
https://www.torahresource.com/radio-files/study-in-galatians/galatians_commentary.pdf