I must admit that the Mathworks explanation is nice here https://www.mathworks.com/help/control/ref/margin.html

I don't agree with the terminology as you now propose.

Gain crossover frequency is where gain is (crosses!) 0dB -- corresponds to gain margin
we can call it Wgco or Wpm

Phase crossover (frequency) is where phase crosses -180 deg -- corresponds to phase margin
We can call it Wpco or wg

Gain margin crossover is simply confusing

Maybe say:
wg: frequency at gain margin (phase crossover point, phase = -180 deg)
wp: frequency at phase margin (gain crossover point, gain = 0dB)