This isn't a full answer, just a sketch.

Well, presumably you want all the elements of $k$ to be units in your local and semi-local rings (ie $k \subseteq R$)? Otherwise things can get very complicated.

I think such a classification does indeed though exist, here's how I would proceed:

Take $R$ to be a local or semi-local (Noetherian?) ring in $F$ that has all elements of $k$ as units. Consider the normalization $S$ of $R$. This normalization should be semi-local and an intersection of those DVRs. Then you can re-obtain $R$ from $S$ by doing a pullback in the category of rings.

See the answers:

Is there a "geometric" intuition underlying the notion of normal varieties?

and

Obtaining non-normal varieties by pushout

The relevant thing is that you noticed that $S = \bigcap R_i$ where the $R_i$s are DVRs, then you would pick ideal an ideal $I \subseteq S$ (which can just be thought of as picking an ideal in each $R_i$) and forming the pullback of the diagram

$$
\{
S = \bigcap R_i \to \prod S/I \leftarrow A
\}
$$
where $A$ is some subring of the (usually Artinian) ring $S/I$.