At NuWho's series three, the Master and the Doctor say they chose their names at a young age. So perhaps, if Hurt's Doctor is pre-Hartnell, he gave up the name to do that thing he did, and then changed into the Hartnell Doctor?
The question is which name, their secret names, Gallifrean names or their current names.
It also interesting that the Time Lords always call him "The Doctor" and not Joebobsmith the Time Lord. If it was pre-Hartnell, then what he did was so awful that even the Time Lords rejected him.