Isn't the use of the word "inside" kind of vague? For instance, I could supply g = -f and the surface would be the same, but the "interior" is now the points outside of the unit sphere.
If we use the function of the plane previously, there really is no notion of "inside" or "outside" either since the function doesn't enclose a region so how would we define it in that case?
Even if inside/outside tests on implicit surfaces are easy, isn't the process of finding the function f that implicitly defines the surface itself hard? Would this make implicitly expressing a surface useless in practice?