Wait.... How am I not a Christian?

I've been wanting to ask this for a while. Every once in a while I will run into someone who says something along the lines of "I used to be Catholic before I became a Christian" or Mormon, or Baptist, or Lutheran, etc... My definition of Christian has always been someone who believes that Jesus Christ was the son of god and that he died for their sins. How are any of those religions not Christian? Is there some new church that calls itself Christian? Is there a Christian church that calls itself that? Or are these "I used to be but am now Christian" people just nuts?