I think many people confuse SP with relaxation or even say SP instead of relaxation.
I suspect you are right about this.

When you are aware of that you are in SP (it's called ASP in clinical literature) you can't move. Well its not quite true...( I know many types and degrees of SP). The point is that you either don't want to move because you are deeply occupied with something (might be remembered later as having been unconscious) or you dream about that you are awake and lying on your bed desperately trying to stop your thought processes.
Well, real sleep paralysis occurs because glycyine (an amino acid and neurotransmitter) is produced in medulla oblongata (the lower part of the brain stem). The signals that control the skeleton muscles pass through medulla oblongata, and the effect of glycine is that the nerve impulses are inhibited. In other words it's entirely a physical effect, and you won't be able to move no matter how much you want to.