Instead of the Hybrid...

[The Wormhole]

Some versions I believe have him being exiled by the time lords instead of simply running away,

IIRC, he's always been running away. If the First Doctor were exiled, than the Second wouldn't have been put on trial for running away.

 

[Solid Snack]

The first Doctor eluded to himself as a sort of exile in the first episode, and seemed to kind of want to get back to his home planet. There's also the "47th century" line in one of the pilots, but given that it's never 100% clear when Gallifrey "is", the latter kind of works.

 

[The Wormhole]

but given that it's never 100% clear when Gallifrey "is", the latter kind of works.

Wasn't there something about it being in the "distant past"?

 

[Solid Snack]

I'm not sure if that's ever in the TV series, but I think it was part of what the novels suggested. Of course as of "Hell Bent", it's now in the distant future.

 

[The Wormhole]

Of course as of "Hell Bent", it's now in the distant future.

They're not really very clear on that. They say they're towards the end of the universe, but the way they talk they make it sound more like "end" as in the edge of the universe, "end of the universe, give or take a solar system."

 

[matthunter]

Then there was also the suggestion that it was outside time altogether, to protect it from any changes to the timeline and allow the Time Lords to take corrective action. And that TARDISes were programmed to only be able to travel to their own native time on Gallifrey to stop anyone mucking around with the Time Lords' own history (to an extent, this was also meant to be why the Doctor and the Master kept meeting in the right order). Of course, we've now seen the Doctor's TARDIS travel back to his own childhood, though you could argue that's because the situation artificially linked the Doctor's timeline with that of Danny (through the toy soldier).