I think "unsprung weight" is a different idea, at least in bike racing. Unsprung weight represents the weight not supported by suspension. In road racing, the suspension is the rider, so the entire bike is unsprung weight.

Not quite. This is a direct relation to "rotating mass", which does include tires and rims but also includes stuff like driveshafts, which are made of shockingly thin tubes of steel or carbon fiber for this reason.

Unsprung mass is relevant to vehicles with suspensions (including mountain bikes). The inertia of the vehicle is set against the inertia of the wheel when it's being pressed back down on the backside of a bump.