There are a lot of inconsistencies in the first season, as Roddenberry & Co. were still settling on things like the name for Starfleet (UESPA, Space Service, etc.), Spock's race name (remember the name Vulcanian in "The Paradise Syndrome"?) and the actual year in which the show took place.
However, I believe they listed 1996 as a definite date for Khan (I can't remember at the moment if that was when he left, or when the war began). Also remember that Kirk didn't know who Khan was when he mentioned 200 years...so, if the DY100 was in service in the 2060s and later (and it's quite possible that DY100s were being used by the recovering Earth after first contact in 2063), then it may have been a legitimate estimate...and one that Khan (who clearly wished to hide his identity at the beginning) wasn't going to dissuade (especially until he was fully aware of his situation).
The Star Trek Chronology, I know, lists the Eugenics Wars as in the last few years of the 20th Century. Who cares if it didn't happen in RL?
Davy Jones
"Frightened? My dear, you are looking at a man who has laughed in the face of death, sneered at doom, and chuckled at catastrophe! I was petrified."
-- The Wizard of Oz