The incarnation of the Gallifreyan played by Sir John Hurt has become known, out-of-universe, as "The War Doctor". However it's unlikely that he was actually called that in-universe. As he made it plain that he should not be called "The Doctor" in that incarnation ("Doctor no more!"), how did other people address him?
We never see the War Doctor addressed directly by his comrades... but Twelve overhears this bit from a soldier in Rassilon's entourage in Hell Bent.
First thing you notice about the Doctor of War is that he is unarmed. For many, it's also the last.
Since this is a Gallifrey that had survived the Time War and not done much else since, hiding away at the end of the universe, it seems pretty clear that the soldier is referring to the incarnation known during the war.
So the only in-universe title we get is the Doctor of War. Presumably since he had given up the name Doctor, he didn't like this appellation, but it seems fame (or infamy) didn't care. Whether anyone else (beyond this one overawed soldier) dared call him this name to his face is probably unknowable at this point.
The Daleks, and the Moment, continue to just call him The Doctor through the events of Day of the Doctor, but as enemies and a somewhat quirky godlike AI, it can be assumed they didn't care for his ideas of what he should be called either.
This is a TIMELINE issue.
The Doctor of War uses the name of the Doctor for the majority of that regeneration.
The Doctor's later incarnations only stop calling him the Doctor at the very END of his life. For those of you saying the War Doctor didn't live long take a look at the reflection in the minisode when the War Doctor is "born." It's at about 6:36-6:39. He starts young.
It's only when he presses the button (or they think he's pressed the button--all those Doctors in one place makes the memory a bit fuzzy, he'd only remember when the latest incarnation experiences it) that the realization that he should not use the name of the Doctor sets in for him. He regenerates into what we call the 9th Doctor pretty much immediately after he "does" this.
Now, as far as Gallifrey is concerned, there's no reason not to call him the Doctor. The Doctor of War is simply a longer title--it's another name for him, that they might use no matter what his face is. It's pretty likely that there may be a number of legendary titles the Doctor is known under--and this is one. But for the most part, they call him Doctor.
As far as the population of Gallifrey is concerned HE WAS NEVER UNWORTHY OF THE NAME. They know he saved them, not that he destroyed them.
Now, as to the title Doctor No More, the first thing he says when he regenerates into the War Doctor--that can be read two different ways, actually--just like the painting "Gallifrey Falls/No More." "No More" is the Doctor's catchphrase in this incarnation. So it could be "I am the Doctor no more" or "Doctor No More," as in "I am the Doctor of No More. I am the Doctor of the line in the sand, the one who will stop it all." We know for sure that he stopped being considered the Doctor by himself once he pushed the button.
Prior to that--if he did ask that others not call him Doctor, we did not see it (except for the Moment!) But even if he did, even as the War Doctor, the soldier states that he did not often use weapons, which tracks more with previous incarnations. I would say there's a difference between his perception of things (namely himself) and how they really are. In the Demon's Run two-parter River calls him out on it. (Armies turn with a word from him). In the Stolen Earth/Journey's End it get pointed out again by the progenitor of the Daleks. (He turns ordinary people into weapons). Clara's boyfriend calls him on it as well. (He's good at being a general.) He's always been able to turn anything into a weapon, always been good at leading people into battle. This incarnation is just a bit better at it--or at least, a bit more able to make the hard choices.
The last big choice the War Doctor makes is the one he's been dreading making--the one that he knows will erase him as worthy of the Doctor's name for certain.
I don't know if the War Doctor stopped Gallifreyans from using his name, at least in front of him, but right up until that last choice he makes, it's still, to quote the Moment "the name in his head." It's fairly likely, that people were still calling him the Doctor because he was worthy still of the name, even if he was fighting in the war.
I'm pretty sure he only interacted with other Gallifreyans (and then only rarely) and his previous incarnations. Most of the Gallifreyan high command didn't need to address him (or if they did, they did it offscreen) and would likely have called him by his 'real' name. His previous incarnations recognized him as a version of themselves, and didn't really address each other.
The War Doctor was not likely a long-lived incarnation - he chose the form of a warrior, did something terrible (which caused his next few incarnations to run, count, and forget), and then likely triggered his next regeneration.
As such, the majority of his 'life' is most likely seen on-screen, and he is essentially not called anything except, "You!"