I'm pretty sure that generally some particles break off from either side whenever you cut something in half. When I cut paper with scissors I get a distinct smell, that's clearly paper particles that have escaped into the air. Under the right conditions you may even see some dust.

When using a saw it is very explicitly removing material to create a gap between the two sides. You can see this clearly in a lot of woodworking videos on YouTube. For other tools like a knife, it's not as obvious, but I still think some material will inevitably be lost no matter what you do.

Maybe some extremely specialized nano-scale methods can cut things without losing material, but I doubt that's something you can do on an everyday life scale.

Disclaimer: I'm not an expert and I did 0 research, just giving my opinion and personal knowledge (which may be wrong).

The term for the material removed to make a cut is kerf. You hear about it a lot with woodworking as not taking into account can lead to inaccuracies in the final dimensions of your materials.

I would think separating a material by passing another through it is cutting while separating materials with only force would be breaking if it happens along a length or tearing of it happens gradually from a single point.

Are you talking molecular level? Then yes, some is lost. If you're talking human noticeable levels, it depends on the material and cutting method.

The CNC (computer controlled cutters) metals that slide into each other so tightly that you cannot see the seam, are created using 2 separate blocks of metal, one becomes the outer block and the other becomes the inner block, then they're both polished together to appear seamless. There's no practical way (as far as I know) to make significant cuts without losing or malforming some of the material (you can cut playdough without losing any of the mass, but it's bunched up along the cut path.

My very old understanding of how materials behave

It will remove some amount of material when you cut.

If it is enough for anyone to care about is another question.

Or was the question what happens on a molecular level and if cutting is any different than sawing at that level?

If you shear (scissors) then there should be less loss, maybe some distortion though. If you saw, you lose the sawdust. if you slice (knife), it might be either way or a bit of both.

Most likely you lose something, whether it's a loss that matters, that depends.

I believe, chemically, molecules with a stronger bond push molecules with a weaker bond apart. And physically, the applied force over the affected area decides the amount and direction of the bonds being separated.