No, this is wrong. It is quite simple, really - same diameter (more precisely, entrance pupil), same light.
You are mistaking light candelas/luxes with lumens. You get the same amount of light in lumens, but not the same light density (luxes/candelas), since the light is spread all over the sensor. The sensor will give you a charge depending on this density of light, and the lens will give you more or less *total* amount of light depending on the aperture (pupil) diameter. Then this light is spread over an area, but a crop lens is designed to focus it on 24mm area, while a full frame lens is designed to cover 35mm... then the 35mm one needs more light entering to yield the same exposure.
For an f2.8, no matter which lens, as far as it's appropriate for the given body, they all produce the same light density on the sensor. Otherwise the exposure settings would need to be different on each body, and you know that's not true. You just cannot compare an f.4 from full frame with an f2.8 on a crop. They do not yield the same photo on the same iso/Tv settings at maximum aperture.
The difference between an f2.8 on a FF and a f2.8 on a crop is that even if they show the same exposure, they are not equivalent in terms of DOF, frame, etc....
But anyway, there's no point discussing what's already discussed elsewhere, so please better read here:http://www.josephjamesphotography.com/equivalence/#exposure