I'd say, at least in the context of American football (and probably real football, too), fast means being able to get from one end of the field to the other in a short period of time. Quick involves being able to do it without being interfered with by opponents.
Yes, very true so. Per consquence this means quick observation, decisions, flexability, an eagle's eye; all things that have to do with keenness, mind.
(btw. some sports clubs here carry names like 'Quick'( my son's football club),
'Quick Steps' and 'Be Quick'. No sports club with the name 'Fast".
