BYU. Are you saying that game isn't a rivalry?
I understand your point about rivalries being competitive, but I think even more so a rivalry comes down to (1) frequency of play, and (2) how much the players and/or fans get up for the game. If the game "means more" than the other games on the schedule and it's a regular opponent, then I guess you could call it a rivalry. Also, a trophy or bragging rights being on the line is bonus points for it being a rivalry.
By those metrics, I'd call Utah St. a secondary rival.