In chapter 2 of "Spaces of PL Manifolds and Categories of Simple Maps" Waldhausen, Jahren, and Rognes show that finite simplicial sets with injective maps as cofibrations and surjective maps with contractible point inverses as weak equivalence, forms a cofibration category.

This category is not saturated. This is because all homotopy equivalences are inverted in the homotopy category since the projection $X \times I \rightarrow X$ is simple, and this forces the two inclusions $X \rightarrow X \times I$ to be equal in the homotopy category. Since there are many homotopy equivalences that are not surjective (not even taking into account Whitehead torsion), there are many maps inverted that are not weak equivalences.

In chapter 2 of "Spaces of PL Manifolds and Categories of Simple Maps" Waldhausen, Jahren, and Rognes show that finite simplicial sets with injective maps as cofibrations and surjective maps with contractible point inverses as weak equivalence, forms a cofibration category.

This category is not saturated. This is because all homotopy equivalences are inverted in the homotopy category since the projection $X \times I \rightarrow X$ is simple, and this forces the two inclusions $X \rightarrow X \times I$ to be equal in the homotopy category. Since there are many homotopy equivalences that are not surjective (not even taking into account Whitehead torsion), there are many maps inverted that are not weak equivalences.