Doctor or Doctor Who? - A clarification about the name of the mad man in a blue box

It has always bothered me a bit, but lately it has come very often to my attention, and I now have the desire to speak out my opinion about this...

People in the Doctor Who fandom always go on about the name of the Doctor. No, I don't mean his actual name, which we will probably never hear, but the name of the character himself on the show. With the following picture/statement probably every Whovian is familiar:

I HAVE ENOUGH OF IT!!!! For god's sake people, don't always get angry when people call the character "Doctor Who" and not "the Doctor". To finally make this thing clear: THE DOCTOR'S ORIGINAL NAME WAS "DOCTOR WHO" - Even Christopher Eccleston's version of the Doctor was still called "Doctor Who" in the credits!!! 

(Sorry for the bad quality)

Basically, calling this man "Doctor Who" isn't wrong at all, it is more of a tradition thing you could say. I just find it very unfair and simply childish to go on about such a little thing which isn't even a real mistake. So I am addressing every Whovian out there: if someone calls this man "Doctor Who" don't get angry because it is not a mistake but homage to the old days.