Paul wrote Galatians in order to defend his gospel against a false gospel that was being handed out in that area.
“Purpose: Paul wrote this letter to defend his gospel against opponents and to dissuade the Galatians from receiving circumcision. Apparently rival missionaries, known as Judaizers, infiltrated the ranks of the Galatian churches during Paul’s absence and stirred up trouble among his Gentile converts. Internal evidence within the letter suggests they preached a false gospel (1:6–7) that pressured Gentile Christians to embrace circumcision and the ceremonial laws of the Old Covenant as indispensable requirements for salvation (5:2–12; 6:12–13). Although they professed to be Christians, they felt that Paul’s gospel of “faith working through love” (5:6) was incomplete without the ritual observances of the Mosaic Law. The success that these Judaizers enjoyed in Galatia forced Paul to respond with a vigorous defense of the gospel (1:11–2:10) and a sophisticated explanation of how the New Covenant inaugurated by Christ dispenses with the ceremonies of the Old (chaps. 3–4). In his view, to add circumcision and other Mosaic requirements to the gospel is to exchange freedom in Christ for spiritual slavery (2:4; 5:1). Stern warnings thus punctuate this letter as Paul appeals to the Galatians to distance themselves from the Judaizers and to disregard their propaganda.” (ICSB)
Given this information, Paul is writing Galatians 2, to give his gospel more authority. He says that he went back to confer with the apostles in Jerusalem and he says “they added nothing to him”. In other words the gospel that Paul was preaching was the same as what the apostles were. This is very important because it means that Paul was not some lone ranger but that he was part of the church that was founded by the apostles. And, it is important in this letter because it means that Paul’s gospel is the same as the apostles giving it an authority that the Judaizers do not have. Whenever we have a dispute we often turn to an authority to settle the matter for us. So it makes sense that Paul would be giving his gospel authority from the apostles that were ‘reputed to be pillars’.
This also means that when Paul mentioned, “James and Cephas and John”, he was talking about the apostles James, Peter and John, who were reputed to be pillars of the church. Who else could be pillars of the church? He does this to establish the authority that he is attempting to give his gospel. So it wouldn’t make sense for him to talk to 3 people who were not apostles to give him authority for his gospel. Those 3 must be the apostles in order for it to make any sense.
Paul mentions Cephas because it is an authoritative name, continuing with the theme of authority. Cephas was a title given to Peter by Jesus. It would be like calling someone the Administrator. If someone disputed with you about something you did in your organization, but if you had permission from the Administrator, you could tell them that the Administrator gave you permission and that would settle the matter, because the Administrator has authority. However, if the Administrator’s name was John and you went to coffee with him, more than likely you would say I went to coffee with John, and not use the term Administrator. Since John is more personal and everyone in your organization knows that he is the Administrator.
Paul may be using both the terms Peter and Cephas in a similar ways. He uses Cephas as term of authority to establish the authority of his gospel. And, he uses Peter, a more personal name, since Paul is close friends with Peter.
Also, when later Paul says that “Cephas stood condemned”, he does so after he establishes that his gospel and that of Cephas, James, and John were the same. So Cephas was condemned not for having the wrong gospel, but for not living up to his own gospel on the matter of gentile circumcision.