Huh, I'd never thought about all of this. I dunno, I always imagined that the player character was in prison by mistake, or destiny had led him/her there. I never though he/she was an actual criminal.
Arena: Jagar Tharn sent the player to prison due to fearing him/her.
Morrowind: The emperor knew you would be the Nerevarine and it is very well possible that you may have been placed in jail by a higher authority or an immortal being, such as Azura.
Oblivion: The CoC doesn't seem to recall why he/she is in jail and isn't even supposed to be in that cell. Uriel Septim said "Perhaps the gods have placed you here so that we may meet."
Your thoughts on it work perfectly fine, but, of course, it's also easy to imagine whatever you want as being the origins, in the case of Morrowind and Oblivion. Arena and Daggerfall give you backgrounds. Arena has you being a member of the Imperial court that was imprisoned by Jagar Tharn and Daggerfall has various backgrounds based on one's class and race.