On the covenants in the initiation of becoming a Christian

It's not the old covenant reinterpreted. The old covenant served its purpose, then in came the new covenant. Paul did not offer a new interpretation of the old, but used what had been written about it centuries earlier to show the foretold change-over had taken place. This new covenant is not a reinterpretation of anything.

As you say that "I am planning on becoming a member of the Church of Scotland" I assume that you seek answers from those in that Protestant and Presbyterian 'stable'. I am a member of the Free Church of Scotland, which is also Presbyerian (and have added that point to your question in order to prevent it being removed).

The old covenant (in the Old Testament) was detailed in the writings of Moses. Yet before that was written down, God's covenant with Noah was understood to have its sign as the rainbow, and applied to every single person descended from Noah as being in that covenant, for all time to come. Then came God's covenant with Abraham, with circumcision as its sign. Eventually, we get to the covenant God made with the descendants of Abraham whom he delivered from Egypt. That covenant was formed at Mount Sinai - God requiring his people to keep their covenant obligations if they were to be blessed by him, and (again) circumcision of all males was the sign. It was a legal covenant, with laws that had to be obeyed.

Later, we read in the Old Testament of God's plan to introduce a new covenant:

"Behold, the days come, saith the Lord, that I will make a new covenant with the house of Israel and with the house of Judah: not according to the covenant that I made with their fathers in the day that I took them by the hand to bring them out of the land of Egypt, which my covenant they brake, although I was an husband unto them, saith the Lord. But this shall be the covenant that I will make with the house of Israel. After these days, saith the Lord, I will put my law in their inward parts, and write it in their hearts, and will be their God. and they shall be my people. ...for I will forgive their iniquity, and I will remember their sin no more." Jeremiah 31:31-34

The old covenant required the people to obey all the written laws which Moses presented to them, after God spoke with him on Mount Sinai, and after giving him tablets on stone, written by the finger of God. The written law shows us what the old covenant was all about. But it was never designed by God to save anyone from their sins! That is why he would later bring in a new covenant that was.

The old law covenant simply exposed all people as sinners unable to keep God's law. That is why the system of sacrifices is spoken of as being a continual reminder of their inability to keep God's law, as well as teaching them the holiness of God - that they needed a mediator to stand between them and God's righteous judgment of them. This is explained by the Christian writer of the letter to Hebrew Christians (not likely to be Paul.) Read this:

"[Christ] is the mediator of a better covenant, which was established upon better promises. For if that first covenant had been faultless, then should have no place been sought for the second. [Then the writer quotes Jer.31:31-33.] ...In that he saith, 'A new covenant', he hath made the first old. Now that which is decayed and waxeth old is ready to vanish away.

"...But Christ being come an high priest of good things to come, by a greater and more perfect tabernacle, not made with hands... neither by the blood of bulls of goats and calves, but by his own blood he entered once into the holy place, having obtained eternal redemption for us... For this cause he is the mediator of of the new testament." (Hebrews 8:6-13 & 9:1-28)

'Testament' is an old-fashioned word for 'covenant'. Now check out all these new testament [covenant] scripture texts (not all of them written by Paul.)

"You could not be justified by the law of Moses" Acts 13:39. "By deeds of the law shall no flesh be justified" Rom. 3:20. "For if they which are of the law be heirs [of God and Christ], faith is made void, and the promise made of none effect, because the law worketh wrath" Rom. 4:14-25. "For you are not under the law but under grace" Rom. 6:14. "Ye also are become dead to the law by the body of Christ. ...For when we were in the flesh, the motions of sin, which were by the law, did work in our members to bring forth fruit unto death" Rom. 7:4-5. "For Christ is the end of the law [to Christian believers] Rom.10:4. "A man is not justified by the works of the law, but by the faith of Jesus Christ: ...for by the works of the law shall no flesh be justified... for if righteousness come by the law, then Christ is dead in vain" Gal. 2:16-21. "But if ye be led by the Spirit, ye are not under the law" Gal. 5:18. "For whosover shall keep the whole law, and yet offend in one point, he is guilty of all" James 2:10.

There are dozens more texts I could cite, all showing that now Christ has come and given the one perfect sacrifice for sin, the law of faith (Rom. 3:27 & 31) is what Christians are under, not the old written law that embodied the old covenant. This is what achieves fulfilment of the prophecy about the new covenant in Jeremiah 31.

Sadly, many people fall between two stools here, because they think they must still do things to show they are worthy of salvation, or at least, to be able to contribute a little bit towards it. They have failed to grasp the immensity of the new covenant, which has replaced the old covenant.

So, no, Paul does not "still regard this covenant as relevant to Christians, it’s just he wants reinterpretation" as you think (in your points after your question). That old covenant was nailed to Christ's cross, at his death (Col. 2:14). That was how it was symbolically taken away. That old covenant handwriting of ordinances of the old covenant was "a shadow of things to come" (vs. 17). Now we have the reality - Christ and the new covenant.