... Within a particular size, the physical dimensions are the same (2820, etc.) and it is the windings ("/xx" number) that changes. What little I know about motors tells me that more windings means more power ...
You have something to unlearn then. Sorry.
Here is some theoryFor a given motor geometry, less the windings, the best-case power capability of the motor is pretty much constant. That's because if you put in more windings you have to use thinner wire, so as the current needed to get a certain torque goes down, the voltage needed to get a certain speed goes up.
Assuming that the windings are packed efficiently into the motor, which is a pretty big assumption, for any given motor speed and torque output its efficiency (and hence the heat it generates) will be constant across different windings and winding styles, and it's ability to dissipate heat will be constant, too.
What does vary is the voltage and current that are needed for that speed and torque point -- if you compare a 12-turn motor against an 8-turn motor, the 12-turn motor will need 2/3 the current and 1 1/2 times the voltage to do the same thing as the 8-turn motor.
Onward to factsThe facts are that, first, that assumption about efficient winding packing isn't always true -- but I suspect it's more true for the model motors we use than the industrial servo motors I've worked with professionally. Second, motor manufacturer's don't seem to be very motivated to compare apples to apples -- they're more interested in making a motor look good for the market segment it addresses, without being accused of false advertising. So your friendly local professional motor-picker gets frustrated reading their marketing foof, because it's hard to unwind what they say into hard engineering data.
There are some details having to do with how efficiently an ESC/battery combination can be used by the motor -- the most efficient case is when the motor is perfectly loaded and using 100% of the pack voltage, but that doesn't work for us because we must have overhead, so we don't run out of poop in the overheads. So you need to choose a motor/ESC/battery combination that gives you enough overhead, without going overboard.
Note that DC motors are very obedient: Two motors with roughly the same Kv, prop, and voltage/ESC setting, will do their damnest to turn the same speed -- the difference is that a great big motor will (probably) be more efficient and will (definitely) be better at dissipating heat. So the little motor may well burn itself up trying to make you happy while the big one loafs along and keeps its cool.
The bottom line isIf you have two motors from the same manufacturer made in the same case with different winds,
there is no free lunch. Given the same prop, a 2826/8 may handle fractionally more or less power than a 2826/12, but the biggest difference will be the best battery voltage (and hence ESC current and wire size).
If you need more power and your motor is already as hot as it can stand then you either need to find a better prop for the motor you have (i.e. spin it faster) and either adjust the battery voltage and ESC or adjust the number of turns, or (far more likely) you just have to bite the bullet and get a bigger motor.