I was not impressed by the heat pump water heater I looked at. Recovery time wasn't very good, and it needed to be in a somewhat warm location. I don't have a basement however, and would have been putting my tank in a crawl space.
The on-demand gas heaters can be really good if there is a high demand for hot water in your home. You would have to have a source of gas and I'm not sure if they are natural gas only or if there are propane versions. They can be expensive, but a direct vent gas heater tank is expensive also, so that should be taken into consideration.
Water heaters do have overtemperature limit switches. It is possible you managed to trip yours with the wood stove, or maybe something esle is wrong. The over heat protector is usually located in with the upper thermostat and element, its a red button that can be pushed in. If it clicks, it probably tripped. If it tripped, then something caused the water or the device to overheat and you should take steps to find out what it was. I usually found State water heaters were bad about having the lower thermostat stick "on" and causing that protector to trip. Replace the lower thermostat and ususally all was good.