I would bet that the thermostat in your water heater is faulty, and has a hysteresis value that is too strong.

The heater is supposed to switch on when the water temperature goes below a certain value, and switch back off when the temperature goes above a certain value. Those temperatures should be close enough, so the temperature is almost constant; but not too close, so the heater does not switch on and off constantly.

In your heater, those two values are way too far apart. When the heater is idle, it allows the water to get warm. When someone takes a shower, the temperature decreases, then it finally switches the heater on. And it does not switch the heater off until the water is hot.

Let’s forget about the delay for now. Unplug the appliance, push the button, and plug the appliance again; if it starts running, all you need is a “smart plug”… if it does not start, things will get more complicated.

Does the doorbell have an ethernet port, and is using PoE? Or did somebody just used an ethernet cable because that is what they had available?