Or to rephrase it: does a knife/scissors/etc just split something or does it remove something?

So, solids aren’t really solid. Like, at an atomic level no two molecules are actually touching. Think of it like a magnet floating above another.

What you cut it with matters too, something insanely sharp like a medical grade obsidian blade wieled with skill is so sharp, it’s just going to separate the molecules. And remove very little material.

Try to cut a piece of paper in half with a baseball bat, and lot of material won’t be attached to either of the two biggest pieces. Because it’s a big blunt object.