To maybe put it more easily, and to bridge the gap between the two usages: the facelift looks like it was made by a professional designer and is thus a professional design. The existing site looks amateur because it appears as if the person who designed it would be unable to charge for their design services.
To argue the opposite would require some kind of weird personification of the design, as in, "this design is a wage-earning participant in its field," which just doesn't comport with how people use "professional" when they apply it to products and especially to designs.