‘The Time of The Doctor’ confirms that the 10th Doctor’s cheating death in ‘Journey’s End’ does indeed count as a regeneration. At the time the 10th Doctor claimed that once the regeneration energy healed his body he diverted the remaining energy into his hand, to stop the process.
We know, however, that a regeneration is not just physical but mental. Might that process already have begun, in tandem with the healing of the body? He might have had the same face but could the 10th and a half Doctor been a different person?
Initially there doesn’t seem much of a change in his character but then it does take time for a new personality to form. The Doctor’s arc for the following year, such as dreading his upcoming regeneration and willingness to accept fate culminating in Time Lord victorious in ‘The Waters of Mars’, could have been signs of a different identity, not just what happens when an incarnation lives too long.
This was the version of the Doctor encountered in ‘The Day of The Doctor’, willing to try and marry the Queen of England as a ruse (and more if his boasts are to be believed). Consider that previously the 10th Doctor had indicated that he couldn’t love a human, yet this version uses romance and seduction as another tool in his armoury.
If a new Doctor did form in ‘Journey’s End’ we have another indicator to his personality in the form of his hand clone. The Doctor claims this double is too dangerous to left on his own because he was willing to commit genocide. That this version was born in battle, full of blood and anger and revenge.
The 10th Doctor claims he isn’t like this anymore but if his personality did form in exactly the same situation (and they were in the TARDIS together when their new life began) then the same applies to him.
The hand clone is left with Rose, so that she can change him for the better, just as she did with the 9th. It is worth noting that the Doctor spends the rest of his adventures (that we see) travelling alone. No one changed him.
This could be explored in your own adventures. Is the 10th and half Doctor in denial about who he is? Is he just playing the role of the 10th Doctor and increasingly feeling like a fraud? If he doesn’t know does he realise that his thoughts and feelings are different now and wonder why?
There is plenty of space for additional adventures for this Doctor to take on new companions. They could temper his personality, just as Rose was supposed to do with her Doctor substitute.
Old companions of the 10th Doctor could meet him and realise that there is something different about him. Can he convince them he is the real deal and not an imposter? Might they make him realise that he is different after all.
The Doctor’s dread at his regeneration makes more sense now we know that his next regeneration would be his last. He might regret his decision to hold on to this incarnation. Compare to the 11th Doctor’s attitude that change is good, that it can be necessarily to change with the times.
By holding on the 10th Doctor has denied himself a new stage of his life. He could have been a whole other person. Now he’ll never get to live that life. He has effectively killed that version of himself.
There is potential in exploring what would have happened if he hadn’t made his decision. You can still link this in with canon since it has been established that time can change. It could be that he did change but something rewrote his timeline. The 10th Doctor might also have regretted his decision and altered things so he does regenerate, which eventually ends with the original history being restored (maybe with the PCs help.)
Would we have at the 11th Doctor as we knew him, played by Matt Smith, or would we have had someone else entirely, born from that moment? Either way has interesting possibilities.
If he does become the 11th Doctor we know his personality would be much different without his crash into Amelia Pond’s garden. Reborn amongst his old companions and with a threat that needed to be dealt what personality would be forged in battle and would he survive? Would this change alter the prediction of his death? Would the 11th Doctor have made the same decision in ‘The End of Time’.
Having a completely different 11th Doctor gives you more freedom, especially if a player wants to take this role. He might experience the same adventures that the 10th Doctor subsequently did or he go in another direction altogether.
In this version the 12th Doctor might have been Matt Smith’s versions, meeting Amelia just as he was destined to do. This allows you to tie things back into the continuity of the show and ensure that important parts of the new mythology, such as River Song and the ‘Fall of Gallifrey’, still happen.
In your own games and with your own Time Lord PC this can be important things to consider if they decide they also wish to hang on to the same face. Things might not be as straightforward as they first thought as they find they are a new person in an old body.
These scenarios can be expanded further, without any additional baggage. You can push it even further with the new incarnation furious at their previous incarnations decision. They could seek out a way to complete the transformation without using up a regeneration to do it.
The PCs might meet a Time Lord PC who has managed to retain the same appearance but whose character continually changes. If they meet several incarnations they could forgiven for believing he has multiple personality disorder. This could get complicated if multiple incarnations meet.