zlacker

In the inner expectation there’s only one choice: you’re conditioning on Y so it’s clear that X is the random variable whose expectation is being taken.

In the outer expectation, there’s only one choice: X no longer exists as a potential random variable (as if it were a local variable in the inner expectation) so the outer expectation must be over Y.

I’m not saying that you’re wrong that those subscripts could be used (they often are) but the meaning of the expression is clear after a little while working with expectations.