As English football came about before American football, surely football is the correct term for the game? The term soccer was, I believe, invented to cure the confusion that Americans got themselves into after inventing a new game with the same name but very different rules.
And, as far as I know, the foot part is not because it involved kicking a ball with a foot, but because it was played on foot. However, the is no definitive answer to this - it's been lost in the mists of time.
Personally football (as in English football) is a MUCH bigger game worldwide, and should therefore have priority (in terms of name) over the American 'version'. Perhaps American football should be called Soccer - somehow a contraction of Association Football - or Amfoot, or Ameriball instead, and solve the silly confusion once and for all.