In the notation for the conditional probability, $p(y|x)$, I believe usually we think of this with $x$ having a fixed value and $y$ varying, such that it integrates to one.
In this case, is it acceptable to instead write it, or think of it, as $p_x(y)$?
I.e. $$ p_x(y) \equiv p(y|x) $$
Somehow I would find this to be more clear than the traditional conditional notation.
This is not the best choice for notation. Such notation is commonly used in two scenarios.
First, you can see people writing $p_X(x)$, where they mean "value returned by probability distribution $p$ of random variable $X$ for $x$". In this case $p(x)$ is a less precise shorthand for $p_X(x)$. So if $X$ is random variable for heights of males and $Y$ is heights of females, then $p_X(y)$ means "what would be the probability of observing female's height $y$ according to the distribution of male heights $p_X$". Using this notation, people write $p_{X|Y}(x)$ to denote $p(X=x|Y)$. You can find many uses of this notation in Wikipedia's article on conditional probability distributions.
Second common usage, is writing $f_\theta(x)$ for "probability density (or probability) returned by probability density (or mass) function $f$ parametrized by $\theta$ for $x$"†. In this case again $\theta$ is a property of $f$. So $f_\theta(x)$ is a different spelling of $f(x; \theta)$.
Moreover, you write
In the notation for the conditional probability, $p(y|x)$, I believe usually we think of this with $x$ having a fixed value and $y$ varying [...]
but notice that $P(X=x|Y=y)$ and $P(X=x|Y)$ are not the same, since in first case you ask about conditioning on fixed value of $Y$, while in the second case you ask about conditioning on random variable.
† - There is nothing special about letter $f$, you can use any other letter in here, $f$ and $p$ are the two most popular choices.
You can use if you’re consistent in your notation (but I don't really suggest). Although widely used, $p(x), p(y), p(x|y),...$ are all abuse of notations. They’re ambigious, e.g. what is $p(2)$? $p$ doesn’t represent the pdf of a specific RV.
A better and fully unambigious way is to use $p_X(x)$, where subscript denotes the RV, and inside parantheses the specific value. This way we know which pdf we use.
Your new notation actually may cause people to think “put $y$ in the pdf of $x$”. If you’ll only read your notes, it’s OK upto some extent.
A bad case when you don’t use subscripts is the convolution of densities by the way: $$\int f_X(x)f_Y(z-x)dx \ \ \text{vs} \ \ \int f(x)f(z-x)dx$$
In the RHS expression, it's totally ambiguous what $f(.)$ represents.
