How are ∈ and ⊂ defined to be relations? I understand a relation to mean, for elements $x\in X$, $y\in Y$ and for subset $R\subset X\times Y$, if $(x,y)\in R$ then $x$ is in the relation $R$ to $y$. But how are $\in$ and $\subset$ defined as relations if they're assumed in the definition of a relation?

Techniquely a relation should be a relation over the sets $X_k$. Even though $\in$ and $\subset$ are called "relations", they aren't really defined in this way. They simply come from definitions and axioms of set theory.
Of course you could try to see if you could define $\in$ and $\subset$ over some sets $X_k$, but then you immediately see that you somehow need "the universe of objects" or "the set of all sets", which aren't sets.
