Protecting the paint
Taking a little paint off
Semantics really
Not really. Two different modes of action. Straight wax bonds to and protects the paint, but contains no abrasive. As a result it doesn't remove paint, whether clear coat or colour coat. It can fill and conceal fine scratches, but it doesn't remove them.
Polish and cutting compound contain abrasive, so they do remove a certain amount of material, and can be used to remove fine scratches by that mechanism.