Let $F$ be a sheaf which is trivial on $X - S$. This means that $j^*F = O_{X-S}^{\oplus n}$ for some $n$, where $j$ is the embedding of $X - S$ into $X$. Note that by Hartogs theorem one has $j_*O_{X-S} = O_X$, so by adjunction one has a morphism $F \to j_*j^*F = O_X^{\oplus n}$ which is an isomorphism on $X - S$. So, this means that such $F$ is a modification of the trivial bundle in the finite number of points. In other words, $Coh(X)_{codim 0}$ is the extenion closed abelian subcategory of $Coh(X)$ generated by $O_X$ and all structure sheaves of points.

Now indeed, as Will Sawin pointed out this category only remembers the dimension of $X$ and the cohomology of its structure sheaf --- given two varieties $X$ and $Y$ with the same dimension and cohomology of the structure sheaves one can construct an equivalence of these categories just by taking $O_X$ to $O_Y$ and choosing a bijection between the sets of points
(and isomorphisms of tangent spaces for the corresponding points).