Paul vs. Cephas (II)

Before looking at the Antioch incident recorded in Galatians 2, we need to sort out some points. Basically, who were they that appeared in Galatians 2:6-10? Let me quote them in full.
2:6 And from those who were reputed to be something (what they were makes no difference to me; God shows no partiality) -- those, I say, who were of repute added nothing to me;
2:7 but on the contrary, when they saw that I had been entrusted with the gospel to the uncircumcised, just as Peter had been entrusted with the gospel to the circumcised 2:8 (for he who worked through Peter for the mission to the circumcised worked through me also for the Gentiles), 2:9 and when they perceived the grace that was given to me, James and Cephas and John, who were reputed to be pillars, gave to me and Barnabas the right hand of fellowship, that we should go to the Gentiles and they to the circumcised; 
2:10 only they would have us remember the poor, which very thing I was eager to do.
Who were "those who were reputed to be something" in 2:6?
Who were the "they" in 2:7? Were they those reputed to be something?
Again who were the "they" in 2:9? James, Cephas and John or those who were reputed to be something?
The best way to harmonize these verses would be to equate the two groups. That is, James, Cephas and John were those reputed to be something. They did not add anything to Paul's teaching when Paul reported to them in Jerusalem (i.e. they approved of his preaching). When these apostles in Jerusalem perceived that God wanted Paul to preach to the Gentiles, just as Peter for the Jews and God had given Paul a lot of grace, James, Cephas and John gave Paul and Barnabas approval, blessings and fellowship (the right hand of fellowship). So, here was the background before the Antioch incident.
But why did Paul use two different names to refer to the same person, Cephas and Peter? When Paul talked about the mission to the Jews, he used Peter. When he talked about the pillars, the important people in the Jerusalem church, he used Cephas. Then, did his readers know that Cephas and Peter were the same person, that they were just different labels to refer to the different missions of the same person? We are not sure.
As for the Antioch incident, here is the text.
2:11 But when Cephas came to Antioch I opposed him to his face, because he stood condemned.
2:12 For before certain men came from James, he ate with the Gentiles; but when they came he drew back and separated himself, fearing the circumcision party.
2:13 And with him the rest of the Jews acted insincerely, so that even Barnabas was carried away by their insincerity.
2:14 But when I saw that they were not straightforward about the truth of the gospel, I said to Cephas before them all, "If you, though a Jew, live like a Gentile and not like a Jew, how can you compel the Gentiles to live like Jews?"
Cephas came to Antioch. At first, he ate with the Gentiles. But when certain men from James came, Cephas separated himself from the Gentiles. Paul scolded Cephas for separating himself from the Gentiles.
Question. When did Paul write the epistle to the Galatians? Did he write it before, or after he had written the first epistle to the Corinthians?
If the letter to the Galatians was written after the letter to the Corinthians, Paul had forgotten to be more considerate. Probably the conscience of these people from James was weak. When Cephas separated himself from the Gentiles, he was trying not to disturb the conscience of James' people. Indeed, Cephas was just doing what Paul advised people to do in 1 Corinthians 10:27-29.
If the letter to the Galatians was written before the letter to the Corinthians, Paul's idea was still evolving and maturing. His criticism of Cephas was immature. By the time he wrote the first epistle to the Corinthians, Paul would accept Cephas' action. Whichever way, Cephas was not wrong according to Paul.
Q.E.D.

My dear Advocate, I thank You for giving us two great pillars of the Church. May we follow their footsteps and continue to bear witness to the good news of Jesus. Amen.