Regarding the Eugenics Wars, let's note the occasional plural. There are plenty of conflicting references to these, and even the introductory episode only vaguely speaks of an "era" of which the 1990s are a part. To limit the Wars (or even the War) to the reign of Khan Singh, 1993-1996, would probably be contrary to the sum total of evidence.
The one outright contradiction among the diverse error is the "Dr Bashir, I Presume" bit about Khan himself only having been created 200 years prior to DS9, not more like 350. There'd be nothing particularly wrong about the Eugenics Wars having continued till that later date, say.
That one line establishes that there was just one five-year mission under Kirk.
The worrisome thing about this is that it sort of also suggests Kirk never spent more than five years out there dealing with V'Ger-like unknowns. Did his entire Starfleet career indeed consist of the TOS five-year mission, plus desk work?
That doesn't ring true at all. Kirk went to strange new worlds in his youth, too (say, "A Private Little War" backstory), even if his earliest years were spent doing instructor work at the Academy.
I guess we have to interpret Kirk's TMP words as "five years out there dealing with unknowns like that in the role of a starship CO", as opposed to "in the role of a starship security officer" or whatever. Which still isn't completely satisfactory, because it suggests Kirk never commanded any starship before TOS. Or then we can narrow it down further, with Kirk saying "five years out there in truly
deep space" which rules out his previous commands of lesser starships or other such spacecraft. But if we keep on doing that, we can basically water down all of Kirk's rambling and remove all power of evidence from the statement. Say, perhaps Kirk only counts his second five-year mission because the first one was such a let-down and doesn't look good in his resume?