FK-This is a tough one for me, as I do agree with you about the problem of reincarnation.
This is a difficult one for me as well. What's interesting in Matthew 17 is that Elijah and Moses both appear to Christ in the transfiguration (Matthew 17:3); yet immediately after the event (Matthew 17:11,12) Christ calls John "Elijah". How could this be if Peter, John and James just saw Elijah? Clearly it could not be reincarnation since they just saw Elijah physically standing with Christ (Peter wanted to construct tents for them). They knew what John the Baptist looked like and this dude was not it. One can only surmise that immedicately following this event that our Lord Jesus was telling His disciples John would be similar in power to Elijah.
What puzzles me is how Peter knew these two people were Moses and Elijah.
That sounds reasonable to me. It seems to me that John the Baptist warranted a standing equal to or greater than any of the "heavy weights".
What puzzles me is how Peter knew these two people were Moses and Elijah.
The only thing I can figure for this is that in verse 3 it says that Jesus, Moses, and Elijah were standing there talking. Perhaps Jesus greeted them by name, and Peter heard.