Water heater problems after water was shut off
Is it a leak or does water drain back from the tank when the main is
off?

If there was no leak before the water was shut off then there is no reason for there to be one as a result of it being shut off. (the pump overheating may(?) have caused problems to plumbing connections)
You will not be able to determine if the pump is still functioning until the water is back on and you can see it pumps water.

Is it a leak or does water drain back from the tank when the main is
off?

In my experience the  municipal water supply pressure  is enough to keep the tank full and will not allow a back flow, but i do not know anything about your particular situation and why you have a pump on your water heater.
You also will need to determine if the water heater itself was damaged from firing up without water in it. (did the service person say anything about that?)