When Star Trek first came out in the 1960s, it was originally just called Star Trek. As it achieved fame and became a franchise with future series and merchandise, it was...only logical...that the very first series would become known as The Original Series.
When exactly did the first Star Trek become known as Star Trek: The Original Series?
While it was called "the original," or, "the original series," or, "the live action series," or, "the real Star Trek" once the animated version came out, it was not abbreviated as "TOS." There was, at that point, no official extension of the name. At that time, Star Trek was expected, among Trekkies/Trekkers, to refer to the 1966 series. If one wanted to specify the Filmation series, "The animated one," or, "the cartoon," or something else was added onto the title, but never as part of the title. The original was just Star Trek and the animated was specified with some phrase that distinguished it.
The animated series had only aired for a short time on Saturday mornings which, in that time, was considered a "kiddie show" time slot, so there were many people who didn't take TAS too seriously. Once ST:TMP came along, though, TAS was beginning to be disowned. (Roddenberry had said he never thought he'd have another chance to continue telling Star Trek stories, so he figured he might as well do the animated, but later said it wasn't really part of the same story. Later, though, that view changed.)
I remember, though, at a convention in the spring or summer of 1988, after the last episodes of the first season of Star Trek: The Next Generation aired, when people were talking about the shows, they would sometimes specify TNG or TOS. From there, people tended to specify ST:TOS or ST:TNG.
I never heard or saw anything like "ST:TOS" until ST:TNG started airing. Yes, people may have used the phrase "The original," and many of us had the same thought. We didn't want to say, "The old one," so we tended to just say, "The original." Many may have even said, "The original series," but people didn't consider it part of the title.
There may be an official designation, from Paramount, that was when TOS was officially added, but I think it started with just general reference and conversation and apparently Paramount seemed okay with it and ST:TOS became the standard.
There's little I can add to Tango's great answer, but I do want to point out The Next Generation Writer's / Director's Guide, a manual written by series creator Gene Roddenberry (with consultation from David Gerrold) and released in March 1987 for potential writers / directors of TNG, which had not yet began airing at the time.
The section on page 5 is titled:
What is changed from the Original Series
At an official level, this a likely origin for the descriptor.
