[quietbritishjim]

I used to hate the way Bayesian ML people used p(...), until I realised that strictly speaking for a conditional variable we ought to be writing: p_X|Y=y(x). The variable is X|Y=y so all that ought to be in the subscript.

It's definitely worthwhile everyone using the full notation at least once so they can get a feel for what's really going on. I've spoken to Bayesian ML professionals who are especially unconfortable with that because it conditions on a zero-probability event (if Y is continuous)... of course p(x|y) does too, they just weren't thinking about it before! And (as I think you're getting at) the appreviated p(x|y) simply throws away information e.g. there's no way to represent the identity p_Y(x)=p_X(x) without adding back some sort of subscript.

But on the other hand p(x|y) is obviously much visually cleaner. If you're writing out a more complex identity and the abbreviated notation isn't ambiguous then it generally communicates the idea much more clearly because there's so much less visual noise.