Good analogy.
I've come up with a thought experiment that has helped me understand the difference between the electric heater and the heat pump. Imagine two rooms, room A and room B, separated by an insulated wall. Both rooms are also insulated from the outside. There is a heat pump in the wall that separates the two rooms. Initially, both rooms have the same temperature.
Now, we want to increase the temperature in room A. So we turn on the heat pump. As the temperature increases in room A, it decreases in room B. But as the temperature difference increases we find that the average temperature of both rooms remains the same. So the electrical energy used by the heat pump is NOT being converted into heat energy.
However, if we have the same setup, except this time we use a space heater to heat room A, we find that as the temperature increases in room A, the temperature in room B remains the same. The average temperature of both rooms increases. So the electrical energy being used by the space heater IS being converted to heat energy.