Throughout, all rings are commutative.
Definition 1. A ring $R$ is coherent if the solutions $\mathbf x=(x_1,\dots,x_n)$ to a linear equation $\mathbf{rx}=0$ are a finitely generated $R$-submodule of $R^{n\times 1}$.
So in categorical terms, $R$ is coherent if for each vector $\mathbf r\in R^n$, the kernel of the map below is finitely generated. $$\breve{\mathbf r}:R^n \rightarrow R,\;\;(x_1,\dots,x_n)\mapsto \sum_i r_ix_i.$$
The definition for modules seems to change perspective. I'll first give it and then elaborate.
Definition 2. An $R$-module $M$ is coherent if for each $\mathbf x\in M^n$, the kernel of the map below is finitely generated. $$\breve{\mathbf x}:R^n \rightarrow M,\;\;(r_1,\dots,r_n)\mapsto \sum_i r_ix_i.$$
This syzygy module (the kernel) is comprised of coordinate vectors that make linear combinations with the $x_i$'s zero. Hence, for modules, coherence does not directly control the solution spaces, but rather the spaces of coordinate vectors which kill of linear combinations with a fixed element.
Now, in the definition of a coherent ring, I'm pretty sure the symmetry in $\mathbf r,\mathbf x$ (which heavily relies on commutativity) means the following definition is indeed equivalent:
Definition 1'. A ring $R$ is coherent if for each vector $\mathbf x\in R^n$, the kernel of the map below is finitely generated. $$\breve{\mathbf x}:R^n \rightarrow R,\;\;(r_1,\dots,r_n)\mapsto \sum_i r_ix_i.$$
So we have these two dual (and equivalent by commutativity) definitions of a coherent ring, and from Definition 1' it's clear that a ring $R$ is coherent iff it's coherent as an $R$-module.
I think it's more reasonable to try and base my intuition around the general definition (for a module), but I don't have any. The equivalence of 1 and 1' for rings seems to be purely formal, and I can't find a way to think of it in terms of actual solution spaces.
In the general case, the elements of syzygy modules are just "linear dependence relations", so coherence tells us that each $n$-tuple of elements in the module really only has finitely many "crucial" linear dependence relations.
Like any finiteness condition, this sounds nice to have, but what does it actually mean geometrically? It must have some geometric interpretation if it pops up in algebraic geometry and several complex variables...
Also, just to be sure - am I right in the equivalence of 1 and 1' or have I made a silly mistake?