I agree.
The original christian concept was more like "life after death", similar to the egyptian belief that there is continuation of life in an underworld, which was common in almost all the european pagan world.
Reincarnation as in, "reborn again" is a concept from the indian region. Problem is people mix the resurrection of Christ with reincarnation, which wouldn't be the same. The case of the resurrection is a special ocasion because, in christian belief, he is the son of God and thus rose again from death. Resurrection would also be a continuation of your life (that was put to a stop by death) whether on this world or in the underworld ("Heaven" for christians), while reincarnation means being born again new and starting over.