I disagree that teams have to be on more equal footing to have a rivalry. According to Miriam Webster,
The word rival most commonly refers to a person or group that tries to defeat or be more successful than another person or group, which means that rivals tend to come in pairs.
I also add in other teams we really don't like. PC will NEVER come close to our level of success, but they are like an annoying little brother who is a jerk and has an inferiority complex. But I would still consider that we have a rivalry with them, partially due to proximity. And for the most part, their fans are insufferable.
Same with SJU. I don't think they have to beat us to become a rival. I don't have any issues with their fans, but am NOT a fan of Pitino. Want to beat his @ss every time we play them. I'd consider that a rivalry.
To me, considering another a team a rival isn't giving them any "power" per se, just acknowledging that as a fan, I don't like them, and want to beat the snot of them every time we play. Like by 40+ points.
I mean I get it. None of these teams are close to the same level as UConn, but rivalries come in all shapes and sizes. Example - on the women's side, Tennessee USED to be equal to UConn, but since Summitt left, they have fallen off the face of the earth. However their fans are the worst in the world, and not enough bad things can happen to that program to suit me. And we rarely play them, but I'd still consider them a rival, just a different kind than ND and SC - 2 current rivals who are every bit as good as UConn...