Pre-Hartnell Doctor - The John Hurt Doctor could have been an awful person, and decided/promised to reform himself, and dubbed himself The Doctor in accordance with that Promise. Then, he went on to betray that promise, until he Regenerated in Hartnell-Doctor.
Let's say you set off on a grand adventure and decide that your new name is Alfred Bestmountfut
but on your adventure you are overcome with emotion and have sex with the Queen's dog. it's your most shameful secret that you are trying to cover up, that you want no-one to discover and you try to make amends for - in your new guise you call yourself.... Alfred Bestmountful
That's the other reason it doesn't work if he was used before Hartnell and it does between Doctors - because he is the Doctor nobody knows by the name The Doctor
so it can be a big event but nobody knows that it was the doctor responsible - which is also why I think it is not the time-war but something just after.