Well, let's see...
Okay, so nope. Instead, he said the era in question was "the era of your last so-called World War". "The Eugenics Wars", as a response to that, could be many things:
1) World War = Eugenics Wars (but it's a mismatch, singular against plural)
2) The era is appropriate for both (sort of trivial)
3) Eugenics Wars is where McCoy homes in, as a specific smaller element of the Last World War era (the likeliest real-world reason for anybody to say such a thing)
4) It's forbidden to call them World Wars now, so McCoy gently corrects Spock, providing both the understanding nod and smile and the politically correct Newspeak name (this excuses the singular/plural mismatch, as Spock isn't mislabeling just the single conflict but the whole concept)
The opposite, basically, as the first three were only the first! (It would be sort of insanely odd for anybody to say "the first three" without there being a fourth at the very least.)
Agreed that calling any of this "intent" is giving the writers (or continuity-minding rewriters and producers) way too much credit. It's just that we can keep on arguing even about the singular intent behind the "Space Seed" dialogue, it already being ambiguous enough.
Basically by the time of ENT, the writers were resigned to doing the fan thing and going both barrels for the fictional continuity of Trek, rather than trying to pretend Trek is our reality where nothing out of the ordinary is allowed to have happened before the airdate (no world wars, that is -
secret visits by bug-eyed or big-eared aliens are fine).
DS9 writing may still have been ambivalent about when Khan ruled, so the "Dr Bashir, I Presume" reference to a "moved" date may be considered either an error or a deliberate re-dating choice. By the time of the ENT stories about the Augments, any reference to non-1990s would be a writing error, though.
The problem sort of being that they had to have been extremely public even back in the day for there to exist this popular reaction of fear and disgust at "80 surviving Napoleons". Loving the treatise to bits otherwise, of course!
Timo Saloniemi