According to the A.D. 30 Jerusalem church, Peter publicly maintained that he had received a personal appearance from the resurrected Jesus.1 This is relevant because they were authorities on whether Peter was actually testifying to this. (After all, he was an authoritative member of that church. Moreover, the other members included the apostles and Mary, the very people who knew Peter best.)
According to the Corinthian church, Peter publicly maintained that he had received a personal appearance from the resurrected Jesus.1 This is relevant because Peter was well-known to the Corinthians.2 If he had not represented himself as receiving a personal appearance, the Corinthians would not have believed it (in fact, they would sooner have believed that he did not receive an appearance).
1 Corinthians 15:1,5 -- Now I make known to you, brethren, the gospel which I preached to you, which also you received, in which also you stand… that He appeared to Cephas [Peter], then to the twelve.
Note: Some scholars, like Bart Ehrman, understandably caution against assuming “this allegiance to Peter was because he… had come to visit the church”, after all “a fourth group, for example, claims allegiance to Jesus himself, and it is certain that he was never there.” [Peter, Paul, and Mary Magdalene (Oxford, 2008), 81.] That said, consider three counter-responses to Ehrman.
―Given Jesus's centrality to Christianity, it is not surprising that such a group claiming Christ in name would rise up without an actual visit from Christ. A “Cephas party”, by contrast, would not be nearly as expected without a visit from Peter who served as teacher or played a founding role for them.
―Paul rhetorically makes special use of Peter's name when saying “Do we not have a right to take along a believing wife, even as the rest of the apostles and the brothers of the Lord and Cephas [Peter]?” (1 Cor 9:5). This emphasizes the high regard in which at least some of them hold Peter, and furthermore suggests that they might be at least moderately aware of his personal life (in this case, that he is married).
―Ehrman doesn't account for the simplest explanation of 1 Corinthians 15:5,8, 11, cited above.
According to Paul, Peter publicly maintained that he had received a personal appearance from the resurrected Jesus.1 This is relevant because Paul knew well whether Peter was claiming this.2
Raymond Brown (NT professor at New York [d. 1998]): “Paul was well informed about the main characters of the Jerusalem church and was in a position to know whether there were traditions that Jesus had appeared both to Peter and James.” [The Virginal Conception and Bodily Resurrection of Jesus (Paulist Press, 1972), 95.]). • Paul had been in a lot of personal contact with Peter. For example, he recounts in Galatians 1:18 -- “I went up to Jerusalem to become acquainted [historēsai] with Cephas, and stayed with him fifteen days…”, and Paul continued to interact with Peter afterwards (Gal 2:9-11, Acts 15, etc.).
According to Luke, Peter had received a personal appearance from the resurrected Jesus.1 This is relevant because Luke likely knew whether Peter agreed to this or not.
Luke 1:1-4 -- “Inasmuch as many have undertaken to compile an account of the things accomplished among us, just as they were handed down to us by those who from the beginning were eyewitnesses and servants of the word, it seemed fitting for me as well, having investigated everything carefully from the beginning, to write it out for you in consecutive order, most excellent Theophilus; so that you may know the exact truth about the things you have been taught.” • Second, consulting eyewitnesses (“living voices”) was expected of ancient biographers. This is a fact recently popularized in Bauckham's award-winning book Jesus and the Eyewitnesses (Wm. B. Eerdmans, 2008). It is relevant because there can be no doubt that Luke's gospel was intended to be taken as historical; scholars fairly unanimously put it as either “Greco-Roman biography” or “historical monograph”. Incidentally, Luke would likely have already encountered Peter in Jerusalem at the conclusion of his trip with Paul, as narrated in Acts 16-21 (Verse 21:17 reads “After we arrived in Jerusalem, the brethren received us gladly. And the following day Paul went in with us to James, and all the elders were present.”). Maurice Casey (Professor of NT literature & language at Nottingham): “Most scholars infer that… Luke was with Paul's party when his narrative uses ‘we’ and not the rest of the time. This is an entirely natural interpretation of a major primary source written by an intermittent eyewitness. Paul also mentions Luke at the close of two or three letters. At the end of the epistle to Philemοn, written in the name of Paul and Timothy when when Paul was in prison, so probably c.62 cε from Rome, he sends greetings from Luke, with Mark, Aristarchus and Demas, referring to them as ‘my co-workers’.” [Jesus of Nazareth (T&T Clark, 2010), 94.]