Fair enough, I suppose. As you say, the industry seems to use the terms assistance, support and amplification interchangeably.For the industry the amplification factor = motor output power / rider's input power.
It is never referenced for the "regular" e-bike legal definition but is often used by manufacturers for marketing purposes. Like, Giant/Yamaha PW-X2 motor of the amplification factor of 360% or Bosch CX with its 380%. It is, for instance, possible to tune the amplification factor of a Giant e-bike by selecting Amplification Factor % directly from 11 preset values spread over 5 assist levels. However, no e-bike I know (except Specialized) offers the capability to set the Peak Power. Therefore, Giant e-bikes always seem to be overpowered and battery hungry.
I was somewhat dejected a few months ago to read in this forum that the 35% boost of my Vado SL in Eco mode was actually more like 63% boost (35% x 1.8) on top of my power input. Perhaps I should reduce the Eco setting to 15-20% to encourage me to put in a bit more effort knowing that the higher boost modes are available if needed?