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Layer via cut is destructive and not recommended. Layer via copy is the better way to go or use a layer mask. I'm not aware of a Photoshop term called "layer cut". I will assume your talking about making a selection and then isolating it using the afore mentioned layer via copy. There's also using the Slice Tool to slice up a layer for something like a web page.

This image is entirely too small. Do you have a much larger version?