Banach-compact operators, $\mathcal A$-precompactness, and frames in Hilbert $C^*$-modules

For a pair of Hilbert $C^*$-modules $\mathcal{M}, \mathcal{N}$ over a $C^*$-algebra $\mathcal{A}$, there are two notions of ``rank-one operators'': $θ_{x,y} : \mathcal{M} \to \mathcal{N}$, $θ_{x,y}(z) = x \langle y, z \rangle$, where $y,z \in \mathcal{M}$, $x \in \mathcal{N}$ (these operators are ca