The only mention of the Enterprises age is in TSFS ("The Enterprise is 20 years old") and it is hard to reconcile with other data points.
Here, I'll admit to having to fudge, but I always though Morrow was talking about the age of the Enterprise
since her refit (which would be fifteen or sixteen years) and rounded up for emphasis.
And I'm no more against a 2230-2235 launch date than I am a 2245 date.
According to the Treknical timeline (as used by Ships of the Star Fleet, Federation Reference Series, etc.) that 20 year figure is taken literally for the time between ST-TMP and ST III... On the surface that may seem like a long gap in years, BUT it really does make sense when you start to take into account a few other bits... Firstly the 22 year duration limit stated in the ST-TMP Blueprints for the Enterprise refit (and restated in many other blueprints)... Morrow's therefore in the right: refit or scrap!
And then we have the Planet of Galactic Peace being established 20 years ago...between the UFP, Klingons, AND Romulans...so that pretty much trashes a 15 year gap between the movies and "Space Seed" in the first season...since nobody even knew what they looked like in "Balance of Terror" (and I don't buy all that behind-the-scenes filler in the Vanguard novels)...