I think the confusion comes from two different things the explanations erroneously lump together as "amount of light".
The true amount of light a lens lets in is only a function of the aperture area. Since area goes with diameter squared, this is proportional to the square of the diameter.
However, what is more relevant to exposure is not the total amount of light a lens can gather, but the brightness of the focused image it produces. This is where focal length gets envolved. Let's say you have a 100 mm lens with a 25 mm diameter aperture (or a adjustable aperture set to 25 mm). Now compare that to a 200 mm lens. If the 200 mm lens also has a 25 mm aperture, then it will let in the same amount of light. However, that same amount of light coming from the subject is now focused twice as large, thereby taking 4 times the area. That means the 200 mm lens with 25 mm aperture creates a image 1/4 as bright (2 f-stops down) compared to the 100 mm lens with the same 25 mm aperture.
Notice that the brightness of the focused image goes down with the square of the focal length, but goes up with the square of the aperture diameter. That means if we were to take the ratio of the two, we'd get a normalized measure of how bright the focused image will be for exposure purposes. That ratio is exactly what f-stops are. These are commonly written as f/n, like f/8.0 or f/11 for example. That is just a expression. The full equation is:
aperture = focallength / n
In the first example of 100 mm lens with 25 mm aperture, that is:
25mm = 100mm / 4
Since that gets cumbersome to write and say all the time and the point is to not have to care what the absolute focal length and aperture are, this is abbreviated to "f/4", with "f" referring to the focal length of the lens and "4" being the ratio of that focal length to the aperture diameter. The second example was:
25mm = 200mm / 8
or "f/8". Other than minor light losses and other subtle effects you can ignore most of the time, one lens with its aperture set to f/8 is going to make the same brightness focused image as another lens at f/8 regardless of focal length. This also explains why long lenses tend to be larger in diameter. A 50 mm lens only needs a 12.5 mm aperture to achieve f/4. A 300 mm lens on the other hand requires a 75 mm diameter aperture to make the same brightness image of the same subject. That means basic physics says a 300 mm lens needs to be at least 3 inches in diameter somehow to achieve f/4.