The definition of a symmetric relation is often given as:
$\forall x,y \in X,\ xRy \Rightarrow yRx$.
is there any reason why it isn't:
$\forall (x,y) \in X^2,\ xRy \Rightarrow yRx$?
to my mind, it makes sense to think about each vertex on a 2D grid with dimensions $|X| \times |X|$ as having a truth value for this statement, and in that instance you would be taking $x$ and $y$ from two different, but identical sets..
I use this definition just as an example, but my question is more generally about how to correctly use notation when selecting more than one element from a set in this way. Sometimes I do see things like $\ldots \in X^4$ written, but I am not sure what the pattern is behind it and when it should be used.
Is this just a convention that some people use, and others don't? or should it only be used when ordered tuples are explicitly needed?
Yes, you are correct: writing $x,y\in X$ is just a shorthand for $x\in X$ and $y\in X$, or equivalently $(x,y)\in X^2$.
If you see written $a,b\in X^2$ that just means $a\in X^2$ and $b\in X^2$; in other words, $a=(x,y)$ and $b=(z,w)$ for some $x,y,z,w\in X$ (there I just used the shorthand ;) ).