I make the distinction this way:
A Hot Rod is a traditional, old-school, hot rodded car. Greased Lightning, Cadzzila, T-buckets, 20s-50s stuff that was modified for speed for drag racing, street racing, moonshine running, etc.
A Hot Rodded vehicle can apply to anything that has been modified to be faster. One is a category of vehicles, the other is a process or action that you do TO a vehicle.
So I agree... I can hot rod a van, but I wouldn't take it to a hot rod show because that implies it is for hot rods, not vehicles that have been hot rodded. Kinda like a square is a rectangle, but a rectangle isn't necessarily a square.
Or to beat a dead metaphor: If you hot rodded a 1949 Dodge in the traditional style, it is a hot rod by the classical definition. If you hot rodded a 1982 C10, it's a hot rodded C10, not a hot rod.