On the etymology of "soccer":
1889, socca, later socker (1891), soccer (1895), originally university slang (with jocular formation -er (3)), from a shortened form of Assoc., abbreviation of association in Football Association (as opposed to Rugby football); compare rugger. An unusual method of formation, but those who did it perhaps shied away from making a name out of the first three letters of Assoc. Compare 1890s English schoolboy slang leccer, from lecture (n.).
To your point, the first college football game in the US was, simultaneously, the first college soccer game because it took place before the split between soccer and rugby, and the rules were those drafted by the Football Association in England. The rugby rules were adopted by Canadian colleges, who went on to play against American colleges, who adopted the rugby rules and added the initial changes that created American football.
In every English-speaking country, whichever of these related games is most popular is simply called “football”.
It’s unclear whether the name derives from kicking the ball with the foot (which happens to a greater or lesser extent in all such games), or from the fact that they’re played on foot (rather than e.g. on horseback).