What if Khan was lying for some reason? Or could he have had some faulty memory because of the long hibernation?
I see no reason to go there. As I said, the Shenzhen Convention and the public's outrage at the idea of genetic engineering (as seen in the news headlines Kore unearthed) make more sense if they come after the Eugenics Wars than before them. I really don't get where Terry Matalas is coming from in his public statements; they seem to clash with the implications of what's actually onscreen.
By the way, it's interesting that we are talking about Eugenic Wars not an Eugenic War, even WW2 is, despite having multiple factions and fronts, only talked about in singular.
Alternatively, could there have been more than one Eugenic War, with the first one ending with Khan going into exile aboard the Botany Bay?
There have been lots of sets of conflicts in history that have been referred to as "Wars" in the plural -- the Wars of the Roses, the Italian Wars, the Napoleonic Wars, the Ottoman-Habsburg Wars, the Navajo Wars, the Opium Wars, etc. What made the World Wars global is that the separate combatants merged into large alliances so that all the local conflicts became parts of a single clash of two main sides. I assume the Eugenics Wars were a set of smaller-scale conflicts that broke out independently when the Augments took power in their respective nations, both between Augments and non-Augments and among the Augments themselves as they jockeyed for power. Spock's description of them as a world war may have been the judgment of history in retrospect.