I much prefer defined characters as such characters feel far more like they are a part of the story and/or a focal point. Custom characters often feel like they are mere tools to further the story and the stories of other characters who feel more like the main characters then your own. The also never really become iconic characters for said particular game or series. Like with say GTA and Saints row. I know who the main characters for GTA are, and they are iconic in a certain way for their respected games, but SR, I dunno, they might as well be random faceless grunt number 1. With say Mass effect I tried a custom looking sheppard, but I just could not get used to it, so I restarted with the default male sheppard with his default name of John, it just felt better that way.
Games like Skyrim/Dragon Age: Origins etc, nice games but your own character, they feel like just tools to further to story of the world and the stories of your companions, who, to me are the real main characters. In MP games it doesn't really matter since you and everybody else is just a random nobody.