It seems like Crossover is just a name for "I can't tell if that's a bulky sedan, a minivan with slightly bigger tires, a small SUV, or a short station wagon." I think statistics that classify them as SUVs are a little misleading because the distinctions are blurring.

It's a platform difference. The starting point for a crossover is a car chassis, whereas SUVs are derived from a truck.

The word crossover implies a unibody car-like vehicle, but people call things SUVs whether or not they are car-based.