1963-1980: Doctor Who and the Early Days
From 1963 to 1980, the main character on the show was listed as "Doctor Who" in the show's credits, and in most of the promotional materials. When people involved in the show were interviewed, they usually called the character "Doctor Who".In-story, the character is almost always just called "The Doctor". There are a few exceptions, but they can all be fan-wanked away. (In "The War Machines", WOTAN demands the presence of "Doctor Who"; in "The Highlanders", the Doctor translates his name to German as "Doktor von Wer"; in "The Underwater Menace", the Doctor signs a message as "Dr W.".)
However, the early comic strips explicitly referred to "Dr Who" in the captions, and sometimes in dialogue.
And the novelizations (which are more important than many newer fans think—for the first few decades, reruns were rare, and there were no home videos, so the novels are the way the older fans remember the old stories) often mixed things up, sometimes calling him "Doctor Who" and sometimes "The Doctor". And most of the titles were Doctor Who and the Foo.
1981-2003: The Doctor and JN-T
John Nathan-Turner took over as producer in 1980, at the end of the Fourth Doctor era. He decided that, since the character is always called "The Doctor" in-story, he should always be called "The Doctor" everywhere else. Starting in 1981, the credits were changed, as was the promotional material. New star Peter Davison and everyone else involved in the show were instructed to use "The Doctor" consistently, and even correct interviewers who got it wrong. The comics also changed to always refer to "The Doctor", and Doctor Who Magazine even had an editorial "reminding" fans that the character is not named "Doctor Who". The novelizations took a bit longer to change over, but they did as well.After the show was canceled, the franchise continued on in novels, comics, and audios. These were mostly written by staffers and fans from the JN-T era, who continued to use "The Doctor" consistently. In fact, the novels even played with the idea that "Dr Who" was a different person from the Doctor—or, rather, two. The Peter Cushing movies, existed as fiction in-universe, and were about a character named Dr Who only loosely based on the Doctor; meanwhile, the Dr Who of the early comics existed in the Land of Fiction.
In 1996, Peter Segal tried to bring the show back with the American TV movie. While some of the early promotional material referred to the character as "Doctor Who", at some point, they switched to using "The Doctor" consistently. This may have been part of his attempt to appeal to the hardcore fans (as when he said that the movie wouldn't have to describe Ace's fate because the New Adventures novels had already done so). The new series didn't happen, so the franchise continued on, as before, in novels, comics, and audios, which continued to refer to "The Doctor".
In 2003, Paul Cornell tried to bring the show back in animated form, with the webcast "Scream of the Shalka" as a pilot. Cornell, who was a hardcore fan from the JN-T era and one of the most popular novelists, also used "The Doctor" consistently. The animated series was announced, and then canceled a few days later in favor of the new series we all know.
2003-2005: The New Doctor Who
In 2003, Russell T Davies convinced the BBC to let him bring the show back to TV, with the first episode to air in 2005.
In the pre-launch marketing blitz, the character was referred to under both names. Stars Christopher Eccleston and Billie Piper called him "Doctor Who" far more often than "The Doctor". When asked about it specifically, rather than saying that the character is named "The Doctor", RTD would give a long explanation that in-universe he's called "The Doctor", but there's nothing wrong with calling him "Doctor Who" out-of-universe.
And when the show was launched, Eccleston was listed in the credits as "Doctor Who".
This is strange—RTD was, like Cornell, a hardcore fan from the JN-T era and one of the novelists. He'd written letters to Doctor Who Magazine about the issue. He'd written characters from other franchises who got into arguments with people who misnamed the character "Doctor Who". Maybe this was his attempt to distance himself from his fan background, like when he claimed that he didn't know much about the novels while a complete set of the New Adventures were visible on a bookshelf in the corner of his office.
2006-2013 The New Doctor
In 2006, David Tennant replaced Eccleston. Tennant was another hardcore fan from the JN-T era, and a participant in the audios and one of the webcasts. He insisted that the credits, and the marketing materials, be changed. He corrected interviewers who asked him about being "the new Doctor Who" (by using the same line as Davison: "Doctor Who is the show, not the character. I'm the new Doctor."). He even corrected Billie Piper on-screen a few times.
In 2010, Steven Moffat took over as show-runner, along with an entirely new production team, and a new cast, starring Matt Smith. At first, Moffat put great emphasis on continuing the established success of the last five years. Also, Smith was primarily a fan of the recent Tennant-era show. So, the character continued to be "The Doctor", although co-stars Karen Gillan, Arthur Darvill, and Alex Kingston did call him "Doctor Who" a few times.
2014- The New Whatever
In 2014, Matt Smith was replaced by Peter Capaldi. Capaldi frequently refers to his character as "Doctor Who". While Capaldi was a big fan growing up, he remembers the early years of the show, and the novelizations, better than the JN-T era.
Moffat has also referred to him as "the new Doctor Who" or "the 12th actor to play Doctor Who". While Moffat is yet another of the hardcore JN-T era fans who'd written (just a short story, not novels) during the wilderness days, he also loves to annoy the kind of fans who like to argue about things like this.
So What Do We Call Him?
RTD's suggestion, that in-universe he's "The Doctor" but there's nothing wrong with calling him "Doctor Who" out-of-universe, may seem a bit hypocritical given RTD's background, makes a lot of sense.
For one thing, to bring up the Doctor in a conversation, it's a lot easier to just say "Doctor Who" than to say "The Doctor, from Doctor Who", or to find a way to reword your sentence so the context is clear in advance, or to hope nobody thinks you're talking about some other Doctor. (For example, when talking about Star Trek, if you say "The Doctor would never do that", people are liable to think you're talking about the Emergency Medical Hologram, aka The Doctor, from Voyager.)
Also, it's hard to argue that the pre-JN-T fans, including Capaldi, are somehow less important than the later fans from the JN-T and Tennant eras.
But still, it somehow feels wrong to call him "Doctor Who". I feel like I'm talking about the Cushing character, or the early comics character escaped again from the Land of Fiction.
Just don't correct other people when they call him "Doctor Who". You always looked like a prat doing that, but even more so now that the star of the show agrees with them.
