....which also shares a common heritage with soccer. Football is apparently a word that was used to describe any ball sport played on foot (as opposed to on a horse). Rugby and soccer are variations on the original rules of football in Britain, according to the original link, and are thus more accurate terms.
American football, on the other hand doesn't appear to share a common link with British rugby, but is just the ruleset for football that eventually became standardized out here.
So from Britain we have soccer and rugby, from the US we just have football.