The word "Fast" in a f/2.8 lens does mean that you can keep a fast shutter-speed for a longer time then with a slow lens like a f/4 it has nothing to do with focus speed, that depends on the camera and the amount of light that is available.
Lens speed refers to the maximum aperture diameter, or minimum f-number, of a photographic lens. A lens with a larger maximum aperture (that is, a smaller minimum f-number) is called a "fast lens" because it delivers more light intensity (illuminance) to the focal plane, achieving the same exposure with a faster shutter speed. A smaller maximum aperture (larger minimum f-number) is "slow" because it delivers less light intensity and requires a slower shutter speed.