I don't think these are the only available conclusions. When I first heard of the theory that Christianity had been influenced by Buddhism - which usually involves some form of the conclusions above - I thought it sounded ridiculous. After getting little bits and pieces here and there (it's a shame I have not found any kind of book that just deals with this matter - do you know of one?), it does seem to me that there is reasonable evidence to conclude:
Some sort of "cross pollination" of the philosophical thought of Buddhism/Hinduism happened with early Christianity. I'm not sure if anyone would have been consciously aware of this, though - it's possible that Buddhist/Hindu thought simply, as I put it before, "cross pollinated" with the philosophies that were already present in the areas where Christianity grew, and since early Christianity seems to have been much better about affirming the truth within the already held beliefs of the people they were trying to evangelize (see Paul at Mars Hill: Acts 17:22-31), it's quite possible that bits and pieces of Buddhist wisdom crept into Christianity (the parallels between sayings of Buddha and sayings of Jesus are undeniable - see "Jesus and Buddha: The Parallel Sayings"). I do not see this as a bad thing.
After Jesus, the legend is that the disciple Thomas (unfortunately labelled with the moniker "Doubting Thomas", even though according to the story, the disciples were all still hiding in a room after witnessing the resurrection and only left after Thomas saw) went to India as the first missionary there. When I first read the gospel of Thomas (which may or may not have been written by him - also, not to be confused with the "Infancy Gospel of Thomas", which is a completely different thing), I was struck with how often the sayings within resembled Buddhist koans. Furthermore, there has been recent scholarship that has felt that this gospel may actually have been used as a source document for the other gospels. So this may explain how thought which resembles Buddhism crept in to Christianity.
EDIT: added italics to one word.
http://www.amazon.com/Jesus-Buddha-The-Parallel-Sayings/dp/1569754616
Check out this book and you'll see that there are many quotes by jesus and buddha that parallel. I like how you said my statement wasn't true without even checking the facts.
And buddhism believes in reincarnation and devas, how is that not faith?