Wikipedia tells us that a lattice is a discrete subgroup of $\mathbb{R}^n$ that spans the whole vector space. This use of the word seems to have its origin in the study of constructions made of laths (OED: narrow strips of wood) that, ignoring boundary behavior, have two directions of discrete translational symmetry.

The original use of lattice has been generalized in several directions. One that is relevant to us is the following: Given a locally compact topological group $G$, a lattice is a discrete subgroup $\Gamma$ such that $G/\Gamma$ has finite volume. In the particular case when $G$ is a vector space over a characteristic $p$ local field like $\mathbb{F}_q((t))$, lattices in this sense are (up to some finite dimensional alteration) a sum of copies of $\mathbb{F}_q[t^{-1}]$.

In the setting of such local fields, or more generally, Tate vector spaces, the word lattice is also used for a sort of dual object, namely a compact subgroup. The two notions are sometimes distinguished with a prefix: $k[t^{-1}]$ is a $d$-lattice (with d for discrete) in $k((t))$, while $k[[t]]$ a $c$-lattice (with c for compact). $d$-lattices and $c$-lattices have finite dimensional intersection, and their sum is finite co-dimension in the total space. See e.g., BBE.

At any rate, if you set $A = k[[t]]$, so $A_t = k((t))$, and $M$ is a finite-dimensional $A_t$-vector space, then a lattice in your sense is the same as a $c$-lattice in the Tate vector space sense. I imagine they were generalizing from this idea.