.... In fact, I think that they must have been referring to two different wars. I believe the Eugenics Wars in "Space Seed" was after the Third World War mentioned in "Bread and Circuses" and thus was the Fourth World War or later in Spock's list of Earth's world wars, and also should have been much bloodier than the Third World War in "Bread and Circuses"....
Which means we have not been very observant. The Eugenics Wars took place 1992-1996, so WWIII was earlier; I have no memory at all of these events. And since Khan appears to be around age 30-35 in Space Seed he would have been alive when that episode was first aired, thus he could have viewed his future self on TV. (Playing fast and loose with known history in this manner kept me from enjoying Space Seed or "The Wreck of Korn".)