Re: So to summarise..
No. Perhaps you should read it again or Google "Rayleigh scattering." If you're sitting at the top of a mountain there's likely to be plenty of discernibly blue atmosphere beneath you, with birds flying through it and whatnot, and with ground beyond it. You probably don't consider that to be "the sky." "The sky" is an effect - a perception: A combination of the optical turbidity of the atmosphere and the innate psychological expectations/interpretations of the observer.
A more reasonable definition of the "sky" perception is something like: "That part of a vista comprising the atmosphere observed in a direction generally upward of horizontal and beyond which nothing terrestrial is discernible from the perspective of the observer"