So would you quite literally call it "Aussie riding"?It depends on where you're from.
I think we'd differentiate between Aussie and Western. To my eyes Aussie is a mix of European and Western and it would be described as Aussie purely because it evolved and is used in Australia. Western would be associated with America.
Here, the majority ride in the style of the first photo and it would be described as 'riding' rather than English. It means that we don't have that split between English and Western.
Perhaps Western would be listed as a discipline or style alongside dressage, showjumping, showing, sidesaddle etc. Most use it for comfort out hacking or competing in western classes.
I don't think we have classes for Aussie riding, although I know a few people who use the saddles