Equivalent Definitions of Types

Trust your intuition! But here is a proof if you don't: suppose there is no $\vec{m}$ in $\mathcal{M}$ realising $\psi (\vec{v})$. Then the same is true in every elementary extension of $\mathcal{M}_A$. In particular, $\{ \psi (\vec{v}) \} \cup \textrm{Th} (\mathcal{M}_A)$ is not satisfiable. (Equivalently, one may assert that every $n$-type in the first sense is realised in an elementary extension of $\mathcal{M}_A$, by Gödel's completeness theorem.)