Probability notation used in ML and engineering has this problem, of overloading p(). Probability notation as used by probabilists in maths departments is completely different: it’s more explicit, and sometimes more clunky.

There’s a hybrid notation that I prefer, for example “Pr_X(x)” for the density function of random variable X at point x; you drop X if the random variable is clear from the context, and you drop x if you’re referring to the entire distribution. Or Pr_X(x|Y=y) for a conditional density. But this notation still has problems when you’re working with hairier conditional distributions, or with distributions that are neither discrete nor continuous.

(Source: used to be a mathematical probabilistic, now working in ML.)