Two Proofs for Open Sets and Metric Subspaces

I have two proofs for the following theorem:

Let $$(S, d)$$ be a metric subspace of $$(M, d)$$, and let $$X$$ be a subset of $$S$$. Then $$X$$ is open in $$S$$ if and only if $$X = A \cap S$$ for some set $$A$$ that is open in $$M$$.

The first proof:

The second proof:

Note that the second proof proves the same theorem but written differently:

Let $$(Y, d)$$ be a metric subspace of $$(X, d)$$. Then $$E \subset Y$$ is open in $$(Y, d)$$ if and only if there exists a set $$G$$ that is open in $$(X, d)$$ such that $$E = G \cap Y$$.

My question is whether these two proofs are equivalent or is one of them more correct than the other? The reason I ask is because of the considerable difference in length! If the first proof is enough, then I would definitely prefer it to the second proof.