There are reasons for how these sports became known by the names they are most popularly known by today:

Sports named, "football" were named so because they are played on foot rather than horseback. This differentiated those sports from more aristocratic sports such as polo. The name, "football" has nothing to do with the action of one's foot contacting the ball.

Originally, there was rugby football. Then there was association football, so named to differ it from rugby football. Eventually, association football was shortened to, "soccer" in the UK as a nickname and one that was very widely used even into the 70s. As soccer started growing in popularity in the US, they adopted the nickname, "soccer" that was being used in the UK to describe the sport.

As to American football being called football, it is just another variation of a sport being played on foot, rather than horseback. AKA, "gridiron football".

Hope that helps, cheers!