I am trying to understand electric circuits (ie voltage, current, power, and resistance). For the most part, everything makes perfect sense, but for some reason I do not feel as if I understand the proper definition of power. Yes, I understand the formula's $V=IR$ and $P=IV$, but I am a strong believer that you do not truly understand something until you can explain it to someone else in layman's terms (which I cannot confidently do at this point in time).
Perhaps I can explain my confusion using the water analogy (which I'm not particularly fond of, but will use anyone for illustrative purposes). I understand that if water is flowing through a hose (or pipe), the amount of water at a specific spot per second is analogous to electrical current, the pressure to voltage, and the width of the pipe to resistance.
Now let's imagine two hoses... one has twice the resistance (meaning it has smaller physical width than the other). But we also make sure that both hoses have the same current (meaning that the smaller hose has twice the voltage (water pressure)).
If we were to hit some toy windmills with the water coming out of each of these hoses (from the same distance of course), it is my understanding that they would start spinning at the same speed, or another way to put this might be that the work done upon them is the same.
Now this is where my confusion starts, because in this situation, the current is the same for both hoses, but the power (watts) is doubled for the hose that requires double the voltage to maintain the same current (due to it having twice the resistance).
When I think of the word "power" and something that has twice as much of it as something else, my mind instantly thinks it is twice as powerful, and thus can exert more force or do more work on external objects. But here it seems like current is what determines the speed at which the windmills would turn, or how bright a light bulb in a closed electrical circuit would be? In fact, it seems like 'power' in this context is a requirement, or the the amount of effort required to keep the current at a constant rate given a certain voltage. This also makes me think that the the hose with twice the power is less efficient (obviously due to the resistance). But thinking of power like this seems counter intuitive, and perhaps I am not understanding something important here. Is power truly the 'effort required' by the circuit to run or is it about 'potential work' that a circuit can exert on external things. Any clarification is greatly appreciated.