Conversion experiences

When I listen to Christian conversion stories, I don't find it at all difficult to believe or understand the part of the story where someone went from believing in no god to believing in a god. In so many stories, people experience a sense of the presence of god. But I rarely understand how that translates into a belief in Christianity, specifically. What about their conversion convinced them that this god they experienced or discovered is the Christian god and that belief in Christ as savior is the only route to knowing God? I'm suspicious that upon sensing the presence of a god, they assume or presume this to be the god with which they're most familiar and this (in the culture of the majority of Americans) is the Christian god, packaged with the associated creeds/dogma of the Christ story. (And yes I have the same suspicions about conversion stories for other religions in other cultures.) I'd like to see people at least consider the possibility that there might be a god, but it may not fit within the confines of the most convenient religion (given their cultural place) or any established religion. If something about their experience is unique to believing in a Christian god, then I'd like to hear about that.