Peak power is the peak of the power waveform. Average power is the time average of the power waveform:
(and "RMS power" is the RMS value of the power waveform, though that's a nonsensical measurement)
(See also peak-to-average power ratio PAPR.)
If you want to talk about an amplifier's ability to remove heat buildup or hold up its voltage rails, that's a separate issue, "peak" has nothing to do with it. That's "continuous" vs "momentary".
True power or Average power is a continuous tone.
When the FTC says "continuous average sine wave power", the "continuous" refers to the long-term tone, and the "average" refers to the time average of the power waveform, as opposed to the peak of the power waveform.
Continuous means the amp is running continuously with the same signal, and any thermal limiting, voltage sag, etc has already happened, and then you measure the average power at some % THD.
You could just as easily measure the "continuous sine wave peak power", which would be measured under the same conditions but would be the peak of the power waveform and would not need an accompanying THD measurement (because it's the same number no matter how much the sine wave is clipped). Peak power is close to twice the average power (depending on the THD of the average power measurement).
Or you could measure the "momentary average sine wave power" or "momentary peak sine wave power", with a burst of sine wave that doesn't allow the supply caps to droop or the heat to build up.
because many companies advertise their speakers and amplifiers with a horrible-sounding squarewave so that the power number is double the real number.
I think it has more to do with what the competition is advertising than any actual measurement.