In ENT "The Augments", Malik states that the Botany Bay was launched at the end of the Great Wars. Just thought this might be worth considering to those who reckon Khan was only involved in the "first Eugenics War."
It's also worth considering that people often misremember their history, or learn it wrong in the first place. It's unwise to assume that every offhand comment made by a character must be strictly accurate and truthful, especially if the character is not a recognized expert in the field under discussion.
I mean, heck, in "Space Seed," Kirk said he estimated Khan had been frozen for 200 years, but it's canonical that the 5-year mission ended in 2270, therefore we know that Kirk's estimate was off by nearly 75 years. (Which makes sense, because he didn't yet know the circumstances of the Botany Bay's launch, and the DY-class ships were in service for decades.)