Q:

What happened to the Doctor at the end of "The Doctor Falls"?

This is the first time I've asked a question here, so I'm going to be extra cautious to avoid spoilers.
In the final episode of Series 10, the Doctor begins the episode with

 a head wound inflicted by the Master and Missy. Then he is knocked unconscious by an electrical shock when a Cyberman grabs him. Later on, his hand begins to glow with regeneration energy, but he forces the regeneration to stop.

Then, at the end, The Doctor is 

 blasted repeatedly by Cybermen and again starts to regenerate, but he refuses to allow it to happen and instead blows up the entire level, destroying the Cybermen.  He delivers his "I hoped there'd be stars" line and closes his eyes.

My question is, at this point, did the Doctor

 die?  It is clearly possible for a Time Lord to die before he is able to regenerate, and it can even happen because the Time Lord refuses to allow the process to happen, as in "Last of the Time Lords," the Master (appeared to) die rather than regenerate just to spite the Doctor.

I'm not sure which is the case. It seems possible that the Doctor did

 die but was restored to life by Bill's tear, the same way that she was restored by Heather's tear. (Bill says that she returned him to the Tardis so that he could "rest in peace," so she seems to think he is dead.) But wouldn't that mean that the Doctor is now a water creature rather than a Time Lord? Or, was he just unconscious, but the regeneration process was still underway?

A:

The Doctor dies (again) and will regenerate
A note on terminology. The question uses "death" to mean death with no ability to regenerate, while the show usually describes each version of the Doctor as dying before regenerating.
It seems clear that (as pointed out in the question) the regeneration process was underway and that the Doctor was trying to prevent it.
The 10th Doctor was in a similar position in Journey's End, but he forestalled the process by transferring the regeneration energy into his severed hand. The 12th Doctor does not have that option.
The question points out that in Last of the Time Lords, the Master appears to choose to not regenerate when he dies. That doesn't seem to be the Doctor's goal in The Doctor Falls. He says that we wants to stay as he is, not that he wants to permanently die.

DOCTOR: Where have you taken me? If you're trying to make a point, I'm not listening. I don't want to change again. Never again! I can't keep on being somebody else. Wherever it is, I'm staying.
chakoteya.net

After the touching scene where Bill says goodbye to the Doctor, the Doctor regains consciousness, leaves the TARDIS and starts to regenerate again (using the snow to halt the process).
It seems clear that the Doctor is fatally injured and will regenerate on December 25th (as advertised).

