But the part I hadn't noticed before is that as soon as the speech ends, Peter dies. There is no mention of punishment, whips, chains or the like. He's just strapped on the cross, gives his speech and then gives up the ghost (another term I had to explain to my son). It is also worth noting that Jesus tells Peter as he is running away from his date with death that the crucifixion represents a second death for Jesus - 'I am about to be crucified afresh.' All of which seems to imply to me at least that the docetic details of Peter's death were shared by the gospel used by the community which produced it.
I strongly suspect the idea derives from 'Simon Magus' (Peter's real name is Simon) who claimed to be Jesus reincarnate. Peter is usually presented as Simon's opponent but it is worth noting that the Quo Vadis ('where are you going') is also found in the Acts of Paul. It has been argued that the story in the Acts of Paul is secondary to its use in the Actus Vercellenses (= Act of Peter), it is not the case of citation or allusion, but rather the adaptation of a narrative unit in a different context.
Carl Schmidt presented the Greek papyrus of the Hamburg Staats- und Universitdtsbibliothek (PH) in his 1936 edition of the Acts of Paul. Its publication solved a scholarly riddle; Origen (Commentary on John, 20:12) attributed the quo vadis scene to the Acts of Paul, but until PH, the quo vadis scene was known only as a component of the Acts of Peter preserved in the Actus Vercellenses. In the Hamburg papyrus, however, the scene appears in the context of Paul's journey from Corinth to Italy. the scene appears in the context of Paul's journey from Corinth to Italy. Jesus walks upon the water toward Paul, who is still on board. He wakes Paul, for it is night. Paul asks him why he is downcast; the Lord responds, "I am about to be crucified afresh.' "God forbid!" responds Paul. Jesus then commands Paul to go to Rome and admonish the Christians and walks before the ship to show the way. Schmidt recognized that Jesus' statement, "I am about to be crucified afresh," was singularly inappropriate as a foreshadowing of the martyrdom of Paul who was beheaded.
There is a complex relationship between the Acts of Peter (in its various forms) and the Acts of Paul. It is generally acknowledged that there is some lost source that is being adapted by both. I strong suspect that this Roman text portrayed another Simon entirely - Simon Magus - or perhaps better yet, that the orthodox caricature of Simon developed from a heretical Peter.