this is the etymology of "soccer" from etymonline.com:
"soccer - 1889, socca, later socker (1891), soccer (1895), originally university slang, from a shortened form of assoc., abbreviation of association in Football Association (as opposed to Rugby football)."
According to this source, at least, it seems it was the British who originally used the term "soccer" to differentiate that code from rugby variants.
All the football variants of the English-speaking world were codified at one time or another during the 19th century.
From my own knowledge, I know that American football was also codified in a collegiate environment, and yes it evolved from the stable of various rugby codes (formal and informal) used in the English speaking world. American football was once heavily dependent on kicking, and grew less so for a number of reasons which I'm sure you guys don't care about.
I will say in the defense of American football that it's much more conducive to a mod because of its abundancy of specialized roles and high number of players (compared to English football). It's also a highly strategic game. But I don't care, because I play Civ for civs, not sports, simulation.
I suspect the use of "football" in Great Britain comes from a Spanish influence. I suspect that Spanish-speakers are the world's majority of the sport and do pronounce and spell it differently than do those of you insisting that your definition of "football" is orthodoxy.