I've been to a ton of Rivalry games from the Iron Bowl (3 times) to Cy-Hawk (9 times) and they are all the same to me.
The fact that a particular rivalry is more meaningful Nationally from time to time is only relevant from a National point of view for rankings in a given year: Alabama-Tennessee (3 times). FSU-Miami (3 times), FSU-Florida (2 times), Oklahoma-Nebraska (1 time), USC-ND (1 time), USC-UCLA (1 time), Florida-Georgia (1 time), etc.
However, when the National spotlight goes away, the fans of the two teams still care very much, and the rest of the CFB world really doesn't care that much.
That being said, to me, the Army-Navy Game (1 time) is the most meaningful individual game Rivalry in CFB that I have seen in person. That is the ONLY game I can think of where the winner TRULY would rather have that win than a GREAT season. Army went 12-2, won the American, won their Bowl Game and the Army QB looked somber after the Bowl Win and could only give the "Go Army" without the "Beat Navy" that usually follows, as it was still too painful to say.
Just my onion.