I don't really know any Star Trek fans, so this problem of nomenclature doesn't really come up. It's the only Trek show my girlfriend is really familiar with, for instance. I just call it Star Trek. Should any further clarification be needed, then I would add "the proper one, you know, with Captain Kirk."
Anyway, what about the Animated Series? The title seen on the screen is also simply Star Trek and that's what IMDb calls it. However, Wikipedia, Memory Alpha, StarTrek.com and TV.com all call it Star Trek: The Animated Series.
Again, if clarification is needed, I call that one "Cartoon Star Trek", which is what the BBC announcers used to call it in the seventies.
That is a good point-the animated series just says "Star Trek" in the title. It seems to me that that was the first time point that new terms were required to differentiate between different Star Trek shows.
Probably where it originated...at the conventions in the early-to-mid 70s when TAS was still on the air and fans had to differentiate between the live action show and the new cartoon one. Acronymspeak probably got its start in the Trekker universe around 1973 or '74.