It's not so much a preference thing for me as it is the simple fact that both methods lead to different playstyles. A pre-defined character is good for a story that's specific and told well (very much not something most devs can do effectively. Yes, narcissist dev reading this post, that probably includes you, especially if your English sucks and you don't read much), but not as helpful for customization or making broadly different plot choices. Loser comes to mind here: there's a lot to do in the game, but all under the aegis of the specific character Nathan. Still, there are things you can do that seriously change how the story and game works because the narrow range of choices you could make allows the dev major story shifts rather than having to return to particular baseline points.
Self-inserting works better for games with a less-fleshed-out narrative that give you much broader character-defining choices. I think here of games like A Spell for All or (the early part at least of) College Daze. In those games, your character has a lot of latitude in what they do, but it's more about locking or unlocking content rather than changing the game's overall narrative direction.
I honestly don't have a preference between them. Either can be done well or poorly, and I've seen a range of them both. So 'I don't care' is as close as I come to a resolution, i.e.: it depends on how well/sexily it's done.