I don't think there's a single "official" moment where they decided it; it's more a gradual transition. The original series premise was deliberately vague about the time frame, since it's hard to predict the rate of technological progress. The original series pitch document said "It could be 1995 or maybe even 2995." And "The Squire of Gothos" was based on the assumption that the show was set in the 28th century (900 years after the age of Napoleon and Alexander Hamilton). But "Tomorrow is Yesterday" and "Space Seed" both assumed the series was about 200 years in the future, and the second-season revision of the writers' bible (the April 17, 1967 edition that's the one generally available) specifies in the "Stardates" section that the date of the series is "actually about two hundred years from now." So that was pretty much locked in by the end of the first year.
However, "Metamorphosis" said that Zefram Cochrane had died 150 years before at the age of 87, meaning he would've been born 237 years earlier. If the series were only 200 years after 1967, then he would've been born in 1930, which seems unlikely. So that kinda nudged the time frame forward somewhat, making it seem likely that it was in at least the early 23rd century.
Later on, the trailers for ST:TMP specified a date in the 23rd century, and Decker said that Voyager 6 -- which would've most likely had to be launched in the 1980s -- was launched "more than three hundred years ago," implying a date in the 2280s or later for the movie -- a decade or so later than we now assume. Then TWOK came along with the opening caption "In the 23rd Century..." as Sir Rhosis mentioned, and the date of 2283 was mentioned for the Romulan ale.
But for a long time, there were two rival theories in fandom about Trek chronology. Some went with the TWOK evidence and assumed that TOS took place exactly 300 years in the future, in 2266-69. But others, notably the authors of the Spaceflight Chronology, tried to reconcile the movies' "23rd century" with the first season's "about 200 years" references by setting TOS in the first decade of the century -- 2207-12 by the SFC's dating system. Which is hard to reconcile with the "Metamorphosis" evidence and the TWOK evidence. The 2283 date for the ale was often speculated to be a Romulan calendar date or a stardate. And the Cochrane issue never seemed to come up; I'm not sure the proponents of the early dating model were aware of the problem. At least, I was an early-dating adherent myself at the time, and I never realized it was an issue.
The first time we got a precise calendar year for any Trek episode or film was in TNG's "The Neutral Zone," which gave the current date as 2364. And that confirmed that the early-dating model was wrong, that TOS and the movies took place in the later part of the 23rd century -- leading to the Star Trek Chronology's codification of the "exactly 300 years later" dating for TOS. I had to redo my whole chronology when that episode aired.