The origin of the confusion is that football (ie: any game that involved kicking a ball) was played in England for about 300 years before anyone thought to put down a set of rules, define how many players on a team, whether it was an offence to pick up the ball and carry it and all that kind of thing.
Because of this, all these games – Aussie Rules, Gridiron, soccer, rugby, etc – are the same game and all have equal claim to the name football.
After all, Aussie Rules football has been played since the mid-1850s which was around the same time as many of the laws of soccer were introduced (free kicks, etc.)
That having been said, I hate the term soccer and think that all other forms of football should have to be spelled out in full; ie: rugby football, Gridiron football, etc.