Suppose I bring a powertool over to a community center (or a relative's home) to volunteer on a project. How can I determine if my tool is getting enough electricity (supposing other loads on the circuit are constant)?
The most basic test: If the tool isn't struggling -- if the motor is maintaining its normal speed under load -- it's getting enough power for practical purposes. If it does run into trouble, taking some load off it by cutting more slowly might help.
But in a normal building, since you aren't going to be travelling with anything huge, the odds of your needing more than 15A are low unless you're running multiple larger tools and/or there are other loads on the same circuit -- in which case, as @jack said, the breaker for that circuit should blow to keep wires from becoming a fire hazard.
(Normally. I've seen some wierdities in buildings with archaic wiring, such as a circuit that dropped from nominal 120V to something closer to 90V whenever someone used the photocopier. Still not sure how that passed inspection, but it did, repeatedly.)
The tool is designed to operate at a particular range of voltages, such as 110-130 V. If the provided voltage is in that range, then it will be getting "enough electricity". If supplied with a lower voltage, the motor could be damaged.
The most direct way to test this is by using a volt meter. I'd suggest using an in-line voltmeter such as the Kill-A-Watt. The tool should be plugged directly into the meter. It will report the voltage being supplied to the tool. You should pay attention to the voltage while the tool is being used. But, tools will usually not use their maximum current unless something weird happens, such as a locked motor. So, while this is a reasonable test, it may not be 100% conclusive. The kill-a-watt will also report the current being used, so that could be used to extrapolate the line voltage during the maximum-current situation.
There is a device to do this measurement for you, perhaps in a safer way: a "power analyzer" device. These present a load (such as 10 amps) momentarily, and report the resistance of your power supply wires. Knowing the resistance and the amount of current your tool uses will let you calculate a voltage drop (multiply them together: V=I*R). Subtract that from your un-loaded power supply voltage to get a minimum expected voltage under load. Ensure this value is in the input range of your tool.
Somewhat related is another question: Will the power supply catch on fire? Fires can be caused by bad electrical connections heating up (or sparking). These conditions cannot be reliably detected, but you can do a few things to minimize the risk: make sure that your supply resistance (as measured by a power analzer) is small and ensure that your circuit breaker works. Note that AFCI circuit breakers also help to guard against fire risk (though I've seen read many opinions only that say they are useless).
The easiest way to tell is by the sound it is making when you are operating it. If it sounds different than normal you can assume something is wrong with the power supply you are trying to use. If the breaker trips when you operate it you can safely assume it is pulling more power than available. Both of these conditions are common to contractors sharing generators and extension cords of dubious quality on jobsites. I have yet to see either of these conditions damage a tool though.
"I'm pretty sure that the voltage at the business end of an under-gauge extension cord would measure the same as the voltage at the wall outlet." You are wrong on this.
If you see a noticeable voltage drop while the cord is under load (tool is on and pulling "max" current) then the wire resistance is too high.
"I've long known that a powertool can be damaged by operating the powertool off of a long, small gauge, say 16 gauge, extension cor" Is also pretty much wrong. Roughly speaking (https://electronics.stackexchange.com/questions/39387/how-are-current-and-voltage-related-to-torque-and-speed-of-a-brushless-motor) speed is related to voltage and torque to current. If Voltage is too low, an internal fan might not be very effective (if it directly geared off the drive of the tool) and the tool will be ineffective. If somehow you can't pull enough current than the tool will be ineffective.
What I suspect you mean by "can be damaged by" is more along the lines of "I drove my tool too hard because I wasn't paying attention or was impatient" for the conditions.
Most hand tools use universal motors. They will run on any voltage up to their rating except that they will run slower and with less power. A long small gauge extension cord just means the tool gets lower voltage and therefore less power. It will not damage the motor. Google and read up on universal motors in hand tools. They can run on either AC or DC and speed control is easy with a variable resistor to change the voltage going to the motor.