errr - I've gotta differ with you a bit.
mAh is the number of millamps that can be delivered for one hour before the cell is entirely dead. Alas, we can't drain them entirely, so the number is somewhat misleading, hence the need for "duty cycle". But if we figure a battery has a 50% duty cycle and 500mAh capacity, that means that it can deliver 250 milliamps continuously for one hour before it's no longer able to deliver adequate current to power the device. Or 500mA for 30mins. Or 125mA for two hours, and so on. Over that period, the voltage will ramp down, but far less for a Li* chemistry battery than for a lead acid or seal lead or nicad, etc.
Not to be confused with the battery's ability to deliver burst power, or it's maximum deliverable current which may be incredibly much higher. Even 30x or more. Although I believe these batteries are 2C max discharge.
So it IS, technically, how "powerful" it is. Noting that power = work/time, and the amount of power we can deliver into a circuit equates to how much work can be accomplished.
Edit to note: And this is the reason I suggest measuring the discharge cycle, because we're dealing with the internal resistance of the battery in this calculation - typically 320-360ohms in a LiIon cell