Hi, Rev Kevin,
On the off change that Benedictus2’s Mother gets the upper hand… let me jump in and offer my thoughts on this. Hopefully, you will not see this as intruding - but, it is an important question, and I really did not want you to wait any longer then necessary…

So, here goes…
Now, in all humility, this question has been asked any number of ways for hundreds of years. The idea that THE idea that answers this question lies in the next few sentences, is brash indeed! What ever happens, ultimately, the Holy Spirit will need to guide the minds and hearts of both readers and writers of these posts.
I think it safe to say that at the time of Christ (approximately, 1AD - 33AD) there were a few recognized religions: Jews and Gentiles would be a major classification - and the Jews were divided into rather lax groups perhaps best identified with Herodians and the really penatential ones best identified with the Esenes of Dead Sea Scroll fame. Gentiles were all over the place - from atheists to polytheists to gnostics to a bunch more - truly all ove the place. About 33AD there appeared this splinter group of Jews who said that Christ was the promised Messiah - and then came the Resurrection and this little group claimed Christ was the Son of God - and this caught everyone’s attention. This attention, however, was intensely focused on their active persection and eradication. One of the most famous of the persecutors was Saul of Tarsus. Now, we do not know ‘why’ or ‘how’ or ‘under what conditions’ but the group apparently called itself “The Way” (instead of Christians) and Saul’s mission was to persecute all who followed “The Way” (Acts 24:4). This apparently met with some success on the part of the persecutors in that these persecuted people fled Jerusalem and Saul went out to Damascus to arrest those who had been previously caught and bring them back to Jerusalem (Acts 9:1-2). It is only later that - for some unexplained and unwritten reason - we really do not hear about The Way anymore - rather, their name seems to have changed to Christian by the believers in Antioch (Acts 11:26). And, I think that is the last time a name for the followers of Christ is mentioned in the Bible. But, you know, it does not end there…
The mere fact that something is not written down does not mean it did not happen or did not exist (John 21:25). So, let’s see if we can go backwards and make some sense of this. Sometime from the start to Christ’s Public Ministry (c. 30AD) and the Resurrection at least some of Christ’s followers said they were members of The Way. This would have been the earliest recorded name for the group. Whether the Apostles said they were members of The Way is impossible to say because nothing is recorded - but, somehow the group, or a group that followed Christ had that name. And, for lack of anything else, let’s just say The Way was the name that Christ’s followers were know by.
Simon was called by Christ to join this group and Christ changed Simon’s name to Peter (John 1:42). Later on God the Father selected Peter and gave him the right answer to Christ question (Matt 16:17) and Christ acknowledged this and gave the keys to the kingdom to Peter and the delegated Power of God to bind and lose anything.
All of this is in Scripture - and requires no further documents then what you already have in your hands. Let’s stop there and see if we are in total agreement or if there is an item that needs clarification.
God bless