As I understand, the F number is a ratio of the focal length to the diameter of the aperture. It doesn't tell you how much light gets through. You could have two lenses with the same max aperture but one could have a much bigger front element that sends more light through it.
You are correct about what the f number represents, as a physical aperture size. However, due to the properties of optics (which I don't fully understand, and certainly can't explain), the f number actually ends up representing the amount of light hitting the film/sensor. Yes, an f/2.8 aperture on a 200mm lens will be roughly 71mm in diameter; physically much larger than an f/2.8 aperture on a 16mm lens (5.7mm) and therefore theoretically admitting much more light. However, due to the way that the light rays bend inside the lens, roughly the same amount of light ends up hitting the film or sensor.
Any given lens at any given f-stop, mounted on the same camera with the same ISO and shutter speed settings, will give the same exposure as any other lens at that same f-stop; no matter the difference in focal length (and thus in the size of the physical aperture).
Let's say we were comparing a 16mm f/2.8 with a 200mm f/2.8. Put the camera on a tripod; take a photo with one lens mounted, then switch out lenses (keeping all exposure settings the same) and take a photo with the other one. The 200mm will of course be "zoomed in" (narrower angle of view) while the 16mm will be "zoomed out" (wider angle of view) but the exposure will stay exactly the same. (plus or minus a small rounding/fudge-factor that is built into all lenses)
That is to say, any given point of unchanging lightness or darkness will appear to be the same degree of light or dark in photos taken with either lens. So a person's face might be tiny in the 16mm photo, and fill the frame in the 200mm photo. But it will be the same level of light or dark. If it is properly exposed with one lens, it will be properly exposed with the other lens. If it is too dark with one, so will it be with the other. So will the sky, and anything else in the photo (save for things like vignetting on the outer edges of the photo -- but the center of the photo would have the same exposure).