OK, point taken. I considered "scaling" in a less general sense (scalar multiple of the unit matrix), while you want to allow arbitrary diagonal entries. My definition is to my knowledge the common one in linear algebra textbooks because in yours, the feasible maps depend on the chosen basis.

EDIT: To state my point more clearly: in textbooks, "scaling" is the linear map that is induced by the "scalar multiplication" in the definition of the vector space (that is why both terms start with "scal").