Identifying an unknown and intermittent source of electric power usage
Find the water heaters relief valve drain pipe and put a bucket under it. if this repeatedly fills with water the cause of the cycling and the lost hot water you are having is from two possible causes.
1. A defective relief valve that is popping open when the water is being heated causing it to be refilled with cold water. Hint to this is when the water to the home is off you have no problem keeping the water hot.
2. The water pressure as it is being heated is rising due to thermal expansion.
I had this issue when I replaced an old failing water heater with a new hi efficiency one. The water pressure coming into the house from the city was 80 psi. With the new water heater I put in problems started due to the fact that water expands when heated and our water meter has a built in check valve to prevent reverse flow.
The pressure would climb until it hit 150 psi and this would pop the safety valve causing it to return to about 90 psi. This would happen every time the water heater ran. The fix was to place a pressure reducer in the line to the house and install an expansion tank.