You are correct, I used to work with batteries for 3 years, people count capacity in Ah or Wh, but never in W. More often in Ah. Why Ah instead of Wh? Because

You are correct, I used to work with batteries for 3 years, people count capacity in Ah or Wh, but never in W. More often in Ah. Why Ah instead of Wh? Because V is constant, making Ah proportional to Wh. Wh = Ah * V; V is the rated voltage the battery is designed for (12V for instance). By definition the capacity is how much energy you can get out of your battery at the rated voltage under standard temperature at a standard discharge rate. Voltage will go down as the battery depletes. When the voltage output is too low and cannot power the circuit anymore we consider the battery empty. Therefore, as we stop using the battery when V is too different, we can consider V constant over the usage period. So Ah = k x Wh, k = 1/ V being almost constant. Ah is a easier unit to manipulate in charge rates, lifecyle etc. One important note: due to technology limitations, energy one can get out of a batter greatly depends on the discharge rate (+-50% energy) and environment temperature (+-50% energy). Capacity cannot be taken literally as how much energy you can store and restitute. Design a grid that can still operate with lower voltage, charge at a lower rate and operate in hot weather and you suddenly have way more capacity (x3).