You'll find there's very little about the Doctor Who canon that's firm and un-contradicted after 50 years of a distinctly laissez-aller approach to continuity. The show's longevity is a result of its ability to morph, chameleon-like, into whatever it needs to be without excessive concern for precedent.
There's no definitive moment that "Doctor" became canon.
But we could say "The beginning of the Fifth Doctor's era" if we go by the credit sequences. It's harder to say when the divide became entrenched in the collective consciousness of the fandom.
Actually, the credits have pretty spotty history.
Like many other of the show's established conventions, the use of "Doctor" instead of "Dr." came rather late (as an example, the idea of regeneration was only invented once they needed to replace Hartnell, and it wasn't called regeneration until the Third Doctor turned into the Fourth).
So you'll find credits which give the character's name as "Doctor Who" or "Dr Who" all the way up to the end of the Fourth Doctor's era, despite his name never actually being "Who" in the show proper.
We can probably blame the ossification of the Dr./Doctor divide on Terry Nation and the fanzines.
Terry Nation invented the Daleks for the show's second televised story, and in doing so probably saved the show from cancellation--or at least from obscurity. But the BBC didn't have the rights to the Daleks, Nation did, and Nation cashed in big time on the popular aliens, with Dalek comics and toys totally unrelated to the BBC's Doctor Who franchise.
He also made Dalek movies, starring Peter Cushing as a human who invented a time machine which took him on adventures fighting against the Daleks. Cushing's character was a PhD with the last name "Who," hence "Dr. Who."
The insistence on calling the Time Lord turned space hobo "Doctor" is partly due to the need for fans (whose vibrant 'zine community of the 70s and 80s still continues alongside and integrated with its online counterparts) to distinguish between the protagonist of the show and the many PhDs he meets, but also to distinguish (and distance) him from Nation's "knock-off brand" Dr. Who films starring Peter Cushing.
The Doctor has many aliases.
The show's name was referred to often as a common in-joke, with the Doctor using aliases like "Doctor von Wer" or "Dr. W," and he's also taken such ridiculous names as "the Great Wizard Quiquaequod" (Latin for "whowhowho"). But the Second Doctor's companion Jamie gave him the alias "John Smith," which he loves to use in New Who. He had a school nickname "Theta Sigma" back on Gallifrey, and he generally picks up names, titles, and aliases like a kid collecting funny-shaped rocks on the beach.
Not every Time Lord takes a title like the Doctor and the Master did. Most of them have "normal" names. Self-appointed titles seem more common among Time Lords who spend a lot of time off-planet, traveling: the Doctor, the Master, the Rani, the Corsair, and so forth. It's been implied that his real name is a string of mathematical symbols and Greek letters, which would fit better with the Cartmel Masterplan notion that the Doctor had an unusually complicated and mysterious past even before he left Gallifrey.