Jesus' appearance to Peter
This must have been an "Oh my God!" moment
The appearances of Jesus after the resurrection are not recorded for us in a consistent form. None of the accounts across the four Gospels agree and the only other reference, by Paul, is given as a "passing on" of "what I have received" (1 Corinthians 15:3). So the reference in the Emmauas story to Jesus appearing to Simon is elusive!
This is a summary of references to post-resurrection appearances of Jesus where Peter could have been involved.
The most significant thing to note is that the women at the tomb were the first to know - not Peter or the rest of the men!
Paul probably wrote his first letter to the Corinthians about the year AD 57 and was therefore the first to recount any of Jesus' resurrection appearances. Paul says that Jesus was seen by Peter, then the twelve (not eleven, as in the gospels), "then by more than 500, most of whom were still alive, then by James and all the apostles". Paul then adds himself to the list (1 Corinthians 15:3-8). However, he does not say just when this was. It is a tradition known to the early Christian community that Paul is passing on. They may well have understood, but we have lost the details in the mists of time.
Mark's Gospel is inconclusive. It was written sometime after Peter's death about AD 65 to 70. In its original form, Jesus did not appear to anyone after his resurrection. It is significant that Mark, who was a disciple of Peter, does not mention Jesus' appearance to Peter. The "Long Ending" (Mark 16:9-20) was added much later to harmonise more or less with Matthew and Luke, and says that Jesus appeared to "the two" on the road (a reference to the story of Emmaus?), but these could have been the two women of Matthew's Gospel or the two men of Luke's. He then appeared to the disciples in the upper room, which would have included Peter.
Mathew's Gospel was written sometime after AD 70 and is in some ways a re-write (with extra bits) of Mark. Matthew's Gospel says that Jesus only appeared to the disciples on a mountain in Galilee (Mt 28:16-20), and once again Peter would have been present.
Luke's Gospel was written later than Mathew, but after AD 70. Luke's Gospel says that on the day of his resurrection Jesus appeared to two disciples walking along the road (Luke 24:13-35). Luke tells us that one of them was called Cleopas (of whom we know nothing) but does not name the other. There is a tradition that it could have been Peter. He then appeared to the disciples in the upper room, on the same day (Luke 24:36-43). There has been some clumsy editing here and the reference to Peter is thrown in almost as a "by the way".
John's Gospel was written much later than the others, around the years AD 90- 100. John's Gospel says that Jesus appeared to ten of the disciples in the upper room on the day of his resurrection (John 20:19-25), then eight days later to all eleven of the remaining disciples (John 20:26-29). He finally appeared to the disciples by the Sea of Galilee (John 21:1-13). Much of this last chapter of John concerns Peter alone (John 21:14-23). Clearly, Peter was present on each occasion. Many commentators regard John chapter 21 as an "appendix", written by John's followers long after his death.
The Gospel writers are telling us the story of Jesus from their own experience, understanding and perspective and it is notoriously difficult, if not impossible, to extract an exact "timeline" of events from them. They recount things that have been remembered by eye witnesses and passed on verbally through the community many years after the events took place. Many "traditions" grew up which we have no means of verifying.