I think sleep paralysis during REM sleep is initiated from the brain, so that you don't act out your dream experiences. The idea of not moving when you wake is all about training your mind not to move so that at a baser level your mind is tricked into thinking that you are still in the sleep paralysis mode. I suggest it's more about not triggering the sequence of events that lead to full wakefullness, which starts with the initial movement.
I think this idea of your "body being asleep" is a bit misleading. I think it's more about the brain shutting itself off from the direct motor control of the bodies muscles, which is still the brain going to sleep.
It's just that there is a closed-loop system where a movement is also sensed back at the brain, it's not a one-way process of your brain saying move and a limb moving.

I know I am terrible at this. As soon as I wake I do a load of stretches, then remember I should be keeping still! I admit though, I haven't practised trying to stay still on waking very much.

Something else to try!