At least in Faltings's setting, $m$ is the maximal ideal of a non-Noetherian valuation domain $V$. If we let $S$ be the non-zero elements of $m$, then the localization of $V$ at $S$ will be the field of fractions $K$ of $V$. The category of $K$-modules (i.e. $K$-vector spaces) is obtained as a Serre quotient of $V$-modules, but one quotients out modules which are killed by some non-zero element of $V$, while the category of amost modules is obtained by quotienting out by a much smaller category, namely the modules which are killed by every element of $m$.
In other words, I think you have misinterpreted the meaning of $m$-torsion, at least in so far it is used in the context of Faltings's "almost commutative algebra".