Is there a difference between $f(x,y)$ $f(x;y)$ and $f(x\mid y)$? While reading I have come across all three of these notations seemingly at random, and as far as I can tell they are all positional arguments to a function, but I can't tell if they mean different things, do they?

Usually $f(x|\theta)$ if $x$ is r.v. and $\theta$ is another random variable (that has a distribution), with known value.
$f(x,\theta)$ is the joint distribution of the two random variables $x$ and $\theta$.
$$ f(x,\theta) = f(x|\theta) f(\theta)$$
Finally, $f(x;\theta)$ is when $x$ is random variable and $\theta $ is a parameter of a pdf, you can also view it as $\theta$ being a random variable with a fixed value equal to $\theta$ with probability 1.
Also more answers can be found in https://stats.stackexchange.com/questions/30825/what-is-the-meaning-of-the-semicolon-in-fx-theta
