Erm, when the rules of the game were introduced as we know them today it was differentiated from other forms of football by being called Association Football .... which was abbreviated to Soccer. They called everything football before that.
Everything is football. Before and after. Look, it's all about what is dominant in your area. If American Football is dominant, then people call that "football" and find another term to use for everything else. If Rugby is dominant, then that is "football" and they find another term for everything else. Same for all varieties of the game: Aussie, Gaelic, whatever. Gridiron dominates in the States, therefore, people invariably use 'soccer' to refer to Assoc. Football. I think a lot of the "it should be called Football!" in the States is just people pushing back on the fact they the game they love (and perhaps prefer) is marginalized by the popularity of the gridiron variety. It's understandable, but it's not going to change anything.
The FA didn't seek to differentiate from other football codes. The whole point was to create one set of agreed rules, which were adopted by everyone. There was a rugby breakaway, but other than that, the agreed rules were adopted nationwide. Other than sketchy references to words similar to soccer for medieval pre-codifed football, the word itself doesn't seem to have been recorded until around 1890, a long time after rugby (and other codes) made themselves known. The fact that football clubs overwhelmingly tended to form as FCs, rather than AFCs (rugby clubs after 1863 were almost always RFCs) hints at plain "football" being the accepted term for the game from very early on, with little need to distinguish itself from the rugby code.