I read that while a relation on X can be defined using the English language, e.g. "a~b means 10|a+b", it can also be defined by the subset of $X^2$ that it produces, whereas I assumed the subset it produces would just be a result of its English definition.
Hence the answer to the question "How many relations exist on the set $\{1,2,3\}$?" is equivalent to "How many subsets exist of the set $\{1,2,3\}^2$?".
But this kind of confuses me, because you could come up with two wildly different English definitions for a relation, but they might end up producing the same subset.
For example, on the set $\{2,3,5\}$, define "a~b" to mean $a+b>10$, and a@b to mean $a^2=b$. Then from what I understand, both of these could also be "defined" by the empty set. So does this mean these two relations are considered identical, mathematically speaking?
Yes, that's exactly what it means. Defining $\sim$ "in the English language" is the same as defining $\{(a,b) \in E^2 \mid a \sim b\}$ (comprehension axiom). In set theory, $a\sim b$ is an abbreviation for $(a,b)\in \sim$.
Now it may seem weird to you that "$a+b>10$" and "$a^2 = b$" define the same relation, if it does it's because you're thinking about $\mathbb{N}$ or something of the sort, where these relations are wildly different, but on $\{2,3,5\}$ they are precisely the same.