I don't think so... basically that would mean every sport could/should be called "football"...
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Basketball = played on foot, not horse therefore football
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Volleyball = played on foot, not horse therefore football
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Baseball = no horse, football again
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Hockey = believe it or not, football... maybe skate ball?
I am kidding of course, it seems nobody really knows and the 2 main theories are the one you mentioned (on foot, not on horse) vs kicking of a ball with your foot (and not your hand)