I am not sure if these formulations of the $m$-equivalence of two structures $\mathfrak{A}$ and $\mathfrak{B}$ are correct.
Two structures $\mathfrak{A}$ and $\mathfrak{B}$ are $m$-equivalent iff
- for every substructure $\mathfrak{a} \subseteq \mathfrak{A}$ with $\left|\mathfrak{a}\right| \le m$ there exists a partial isomorphism $\pi$ from $\mathfrak{A}$ to $\mathfrak{B}$ with $\operatorname{def}(\pi) = \mathfrak{a}$, and
for every substructure $\mathfrak{b} \subseteq \mathfrak{B}$ with $\left|\mathfrak{b}\right| \le m$ there exists a partial isomorphism $\pi$ from $\mathfrak{B}$ to $\mathfrak{A}$ with $\operatorname{def}(\pi) = \mathfrak{b}$; or equivalently - for every substructure $\mathfrak{a} \subseteq \mathfrak{A}$ with $\left|\mathfrak{a}\right| \le m$ there exists an isomorphic substructure $\mathfrak{B} \supseteq \mathfrak{b} \cong \mathfrak{a}$, and
for every substructure $\mathfrak{b} \subseteq \mathfrak{B}$ with $\left|\mathfrak{b}\right| \le m$ there exists an isomorphic substructure $\mathfrak{A} \supseteq \mathfrak{a} \cong \mathfrak{b}$
For reference, the definition of $m$-equivalence I was taught was that for every predicate logic formula with at most $m$ nested quantifiers $\phi$ it is the case that $\mathfrak{A} \vDash \phi$ iff $\mathfrak{B} \vDash \phi$.
Are these three formulations the same?
No. Your formulations are equivalent to the statement that for every sentence $\varphi$ with at most $n$ quantifiers all of which have the same type, $\mathfrak{A}\models \varphi$ iff $\mathfrak{B}\models \varphi$.
For example, $(\mathbb{Z},<)$ and $(\mathbb{N},<)$ are "$m$-equivalent" for all $m$ by your modified formulations (since $\mathbb{Z}$ and $\mathbb{N}$ have the same finite substructures up to isomorphism, namely all finite linear orders), but actually these structures are not even $2$-equivalent, since $\mathbb{Z}\models \forall x\exists y\, (y<x)$, but $\mathbb{N}\not\models \forall x\exists y\,(y<x)$.
An equivalent formulation of $m$-equivalence in terms of partial isomorphisms is that Player II (aka Duplicator) has a winning strategy in the Ehrenfeucht-Fraïssé game $G_m(\mathfrak{A},\mathfrak{B})$.