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.
This story lives in my head forever as a perfect example of everything pointing to a theoretical answer but reality not caring.