How to empty hot water tank when closing up camp?

I have searched the forums and looked at the diagrams (very helpful, thank you!) but am still mystified by the hot water tank.

The problem is that I don't want to run the water pump "dry" in order to push out the rest of the hot water, after being hooked to city water or when draining the freshwater tank in the event it empties out before the hot water does.

I can't find anything on running the water pump without water flowing through it; but there is a little voice in my head saying "you're going to burn up the pump if you do that."

True?

I'm familiar with the relief valve on the outside of the tank, but that isn't what I'm after, particularly when I know it's full of hot water...

I have to admit I'm also mystified by how the hot water tank fills up without the water pump - when you hook to city water does it immediately fill up that tank? It seemed to, but that doesn't make sense since you'd think I'd have to open the hot taps to create the pull to fill it. Needless to say, I never use the electric heating element because I am never sure if there is water in the tank.