The materials library is very useful but there’s one thing that’s counterintuitive and frustrating which is the material/cut name. When creating or editing a cut in the library, you can assign it a name like you can with any cut. However, when you assign this cut from the library to a layer, the name does not come with it. Instead, the current layer/cut name remains and you have to manually edit the cut from the Cuts/Layers menu.
The cut name from the library should be applied just as all of the other data from the cut is applied, or at least this should be an option in case some people like the current behavior.