When to use which quantifier with predicate logic?

The two answers are equivalent.

"$\lnot \forall$" is the same as "$\exists \lnot$".

If not all cats are black, there must be some cat that is not black.

Thus, we have that $$ ¬∀x \ (Gx \to Lxx) \iff ∃x \ ¬(Gx \to Lxx) \text{.} $$

Now we apply the tautological equivalence $$ \lnot (p \to q) \iff (p \land \lnot q) $$

(We can check it with a truth-table: $\lnot (p \to q)$ is TRUE just in case when $p$ is TRUE and $q$ is FALSE.) to get the final result, $$ ∃x \ (Gx \land \lnot Lxx) \text{.} $$

Both are correct. They are equivalent.

$\neg \forall x~(Gx\to Lxx)$ "Not all girls love themselves."

$\exists x~(Gx\wedge \neg Lxx)$ "Some girls don't love themselves."