The biggest difference is from 1998ish on the gold standard for recognizing National Championships was the BCS. While the AP still awards NCs, the only one that mattered through last year was the BCS.I have no problem with their 2004 claim. They were undefeated and never got to play for it on the field. They lost this game on the field. It doesn't matter who played, they lost, and it was not their only loss on the season.
The debates of the past will always be and we rarely got a true #1 vs. #2 like college football got with Bama and Penn State in 1979 or Bama and Miami in 1992 in the traditional bowl system.
But since 1998 at least there was an effort to pit #1 vs. #2 annually. Now, going forward, it'll be the college football playoff system.
FWIW, our only NC that Bama claims that is seriously questionable is 1941, but that is a different debate in a different era.