Trying to determine equivalence of two definitions of multiplication for integers modulo $n$

No it isn't. Working mod $5$, we get $\overline 2\cdot\overline 2=\overline 4=\{5k+4\mid k\in\mathbb Z\}$.

So $19\in\overline 4$, but $19\not\in\{xy\mid x,y\in\overline 2\}$, since the only ways to express $19$ as a product are $1\cdot19$ and $(-1)\cdot(-19)$, and none of these factors are in $\overline 2$.

No, in general your $\overline{a}*\overline{b}$ is only a subset of $\overline{ab}$, and the inclusion can be strict. Think of the square of the zero element of $\Bbb Z/2\Bbb Z$: multiplying two even numbers together always produces a multiple of $2^2=4$, and not all even numbers have this property. Note that the same counterexample works in $\Bbb Z/n\Bbb Z$ for any integer $n>1$, simply replacing $2$ by$~n$.