Why there is no such thing as RMS or watts RMS?

There is such a thing as "RMS power", but it's not useful for anything, so don't use the term. No one measures the RMS of the power waveform. What they do is measure the RMS of a voltage waveform, and then use that to derive the averagepower. The correct term is "average power", not "RMS power". You could measure the RMS of the power waveform instead of the average, but your measurement would be 1.2 times too high.