In Australia, they play "British rules football," "American rules football," and "Australian rules football."
I dunno how they keep it all straight, and my information is third hand and decades old, so they might have different terms by now for all I know.
Oh, it's far more complicated than that.
Which game gets called 'football' varies by state, and to a degree by fandom. In Victoria, Tasmania, South Australia and the Northern Territory, 'football' usually means 'Australian Rules Football', aka AFL or (historically or as a derogatory term, VFL), from the acronym of the governing body, the former Victorian Football Leage, now Australian Football League.
In Queensland and New South Wales, 'football' usually means 'Rugby League Football'.
In Western Australia, 'football' could mean either; or it could mean Association Football - usually known as soccer (I have never heard it called 'British Football'). Mostly it's AFL in WA; But Perth has a much larger soccer fandom than the other states.
Another commonly played code is Rugby Union Football, which (to avoid confusion with Rugby League Football) is called 'Rugby'.
The AFL also arranges matches called 'International Rules Football', which is a hybrid game with some characteristics of AFL, and some of Gaelic Football; Such matches are typically played between an Australian and an Irish team, but do not use the same rules as matches played in either country, so International Rules is a game in its own right.
We also have a number of American Rules Football teams over here, though that code is still a very minor part of the sporting landscape.
Any and all of these might be referred to as 'footy'.