I'm thinking his John Hurt first incarnation he must have done something really really really bad. It makes sense now why he'd want to run a way in a TARDIS and go explore the universe and help people if he did something really bad in his first incarnation, he's been trying to redeem himself all this time. Everything fits into place with this theory.
Well, not everything, although I do prefer that variant over John Hurt being the Time War Doctor - I'll get back to that in a minute. If the Doctor started out as someone who did something incredibly horrible it throws quite a shadow over the more light-hearted 2nd, 3rd and 4th Doctors. Three had his darker sides - his bitterness about his exile on Earth and the will to leave everyone behind at a whim if he could have regained his ability to time travel - but all things considered, he was an eccentric traveller with little regard for personal property. His dialogue with the Master about how all he wants is to see the universe would ring rather hollow. Even worse for all the joking lines about his days at the academy in the old show. Instead of making fun of him for his low grade in the exam wouldn't Romana be more inclined to say, "well, you know, at least I'm not a murdering maniac"?
John Hurt being the Time War Doctor doesn't make much sense to me because for all their emotional trauma neither Nine nor Ten had trouble acknowledging what they did during the War. Nine tells Rose about it soon after they meet, and Ten even admits in "The End of Time" that locking away the Timelords was a deliberate choice for what they had done and then he does the exact same thing again, despite Susan (or whoever she was supposed to be) looking him right in the face. I don't see how a disowned, forgotten incarnation fits into the picture here. Which is probably why they'd go for that route.