I missed some discussion about the Eugenics Wars and WWIII above...
Here is what we know from "cannnnon."
TOS "Space Seed" did lump Earth's "last so-called World War" with "The Eugenics Wars," setting this in the 1990s.
The direct connection between the last World War and The Eugenics Wars was not brought up again in the TOS-verse or in Berman-era Trek.
The movie First Contact placed the end of WWIII at 2053, and then the Phoenix launch happened in 2063. No mention was made of "Eugenics Wars" or anything about genetically-engineered superhumans, though. Supposedly, meeting the Vulcans ushered in this new era of peace and whatnot for all of humanity.
But... TNG "Encounter at Farpoint" had the "post-atomic horror" in 2079, but did not actually give a name to whatever war/conflict led to it. Was it lingering effects from WWIII, or did something else happen after first contact with the Vulcans?
And then SNW pushed some events further along in time. S1 E01 showed that footage of unrest on Earth in the early 21st century, looking very much like the real-world 2020s. Pike says, "We called it the Second Civil War, then the Eugenics War, and finally just World War III." Apparently we are now meant to understand this as a protracted period of conflicts going on for 20+ years, and starting in the 2020s instead of the 1990s.
SNW S2 "Tomorrow..." confirmed that the timeline change was the result of some timey-wimey stuff with "temporal wars." Sera says it was originally supposed to all start in the 1990s, but she's been stuck there for thirty years trying to make it happen.
Edit to reply to another comment:
But still the same "organization" that the Enterprise encounters in Arena, right? Who attack and behave (tactically) pretty much like the Gorn from ten years previous?
Yes, that's my thought; along the lines of the "Gorn Hegemony" from the old Trek tie-in stuff. Different types of Gorn all within one big geopolitical entity or something.
Kor