If you wish to simplify it down to relatively meaningless numbers then you are right, it's not too complicated.
All of the older companies did use the term "RMS" power because that was the EXACT term used by the standards organization at the time, based on the RMS voltage and a resistive load. P(RMS)=V(RMS)squared/R, leaving the terms Watts and RMS in the units. This was a somewhat simplistic use of untis, but served to characterize that the power was based on RMS measurements and not peak or average. Here is an example of the required marking (including the term RMS) by the CEA: Invalid Link Removed (from the new mobile audio standard book). This term was also required IIRC by the FTC.
Note that my involvement in this industry goes back to 1978, when "RMS power" was indeed interchangable (and required to be used) with "continuous average power", as are companies such as Crown, QSC, etc. There have been a lot of different regulations passed around between the different markets (mobile audio, consumer, MI, pro audio) trying to address this, and each market has ended up with what "works" (yes, a loose definition) for them
Over a period of time, the convention "RMS Power" was taken to be based on RMS voltage and current measurements and the term "average power", and "contnuous average power" began to replace RMS except where required by law. Now the term continuous has taken on different meanings at it is a number that means different things to different markets and applications. For an application that has high dynamics (say a full range signal), the peak to average ratio of a reasonable and typical signal is what resulted in the 1/8-power testing that has been almost universally adopted by all national standards (UL, IEC, FCC, etc.) and an amplifier must be capable of continuous operation at 1/8 of it's rated power (based on RMS voltage), so an amp that has say a 100 watt 1/8-power certification is presumed to be also rated at 800 watts continuous average power based on RMS voltage. This may or may not be accurate depending entirely on the peak to average ratio of the typical signal being used in the marketplace and the true load impedance. My feelings are that for a full range signal, or the mid and HF sections of pro audio it's pretty reasonable until you get to pretty heavily compressed program material, but for low frequency sections which are often more compressed by their nature, and for bass guitar applications 1/8-power is inadequate and a more rigerous value should be used (which I do) based on this market's specific signal profile and needs.
Here's a brief look at the topic: Invalid Link Removed