Decker does give us that voyager six was launched more than three hundred years ago, and Kirk indicates it was from the late 20th century, which could easily mean that TMP is set in the early 24th century.
Agreed. Only the TMP trailers mention the 23rd century.
We covered the TOS date references in the other thread, so I'm not sure this thread is needed. "The Squire of Gothos" suggests the 28th century (at least 900 years after the death of Alexander Hamilton in the early 1800s). "Tomorrow is Yesterday" suggests the 22nd. "Space Seed" suggests the end of the 22nd, give or take. "Metamorphosis" takes place c. 235 years after Zefram Cochrane was born (he was lost 150 years before at age 85), so it's most likely about 200 years after warp drive was invented, give or take a decade. Now, Marla in "Space Seed" said that sleeper ships were needed until about 2018, so maybe the idea at the time was for that to be when warp drive was invented, putting TOS in the early 23rd century.
The only dating reference I can think of in TAS is in "The Infinite Vulcan," which echoes "Space Seed" in referencing the Eugenics Wars as occurring 200 years before, and says that Keniclius would be over 250 if he were alive. Yet it's contradictory, because Keniclius has supposedly been out of touch for two centuries, yet is aware of "the galactic wars" involving the Romulans, Klingons, and Kzinti. It implies that the Eugenics Wars came much later than "Space Seed" indicated.