I suggest making it explicit that P is a distribution over a (possibly infinite) set of variables (or propositions naming symbols, or whatever your preferred formalization is), and that P(x) is shorthand for P(X=x) when X is unambiguous. This is one of those things that I had to figure out myself, which had confused me historically in my youth, and led me to think that all the P notation was probably informal argument rather than formal math.
I suggest making it explicit that P is a distribution over a (possibly infinite) set of variables (or propositions naming symbols, or whatever your preferred formalization is), and that P(x) is shorthand for P(X=x) when X is unambiguous. This is one of those things that I had to figure out myself, which had confused me historically in my youth, and led me to think that all the P notation was probably informal argument rather than formal math.