The word 'rival' is more about semantics than anything. By definition, a 'rival' is either a person or thing that tries to defeat or be more successful than another, so it's easy to say that every team on the schedule is a 'rival' when examined under that lens.
Rival is also defined as something or someone that is as good or almost as good as another person or thing. To determine that, are you looking from a historical perspective or more recent?
In terms of historical match-ups, scUM is clearly a rival because of both on the field competition and in recruitment of student-athletes over time. Being able to recruit Ohio products to play for TTUN and our staff now doing their best to bring 'northerners' to tOSU has certainly added fuel to the fire we call 'The Game.'.
No other school, from a historical perspective, is really what you might call a rival, IMO. Though I do not have records right in front of me nor plan to look them up, our Buckeyes defeated Wisconsin something like 62 of the first 65 times games were played against one another. It wasn't until Barry Alvarez got there close to 25 years ago that things changed. Bert, being Bert, added some flavor to the games but he only defeated tOSU once.
As for MSU, I just don't see enough historical match-ups or recruiting battles to really consider Sparty a traditional rival though see this one evolving. Yes, there have been some memorable, or not so memorable, games between our schools (MSU in 1998 immediately comes to mind) but not many historically speaking. Coach Dantonio, being from Ohio and on the 2002 National Championship team, has certainly brought MSU more into the discussion of a rival with Sparty's win this year and the battle for Jamarco Jones.
With no disrespect to PSU, I just don't see them as a true 'rival' based on games played over time nor have I seen many epic recruiting battles. PSU has been very good historically and the games played against OSU since joining the B1G have, for the most part, been highly competitive. My age may have something to do with seeing PSU as I do, as the first 3 games were played in the 1970's as I was closing in on 20. I never had an attachment or any true emotion to this game.
I was taught to respect all opponents but know from both playing and coaching that only certain teams can get the blood flowing. For this Buckeye, scUM is the only one right now that meets that standard, with MSU now in the discussion. Go Bucks!