I think there's an element of expectation-setting when we're talking about authenticity. Personally, I wouldn't sweat authenticity too much. There's excellent food to be had by mixing and remixing dishes.
So chicken sandwiches, pork sandwiches, and steak sandwiches, which are all common here and typically served on buns, aren't considered burgers because the meat isn't ground.
It's so thoroughly thought of this way here that that the product "hamburger helper" doesn't involve bread or sandwiches whatsoever: it's a box of pre-portioned ingredients, maybe with pasta or rice, that you cook with with ground beef in a pot or skillet.
Edit: old timey American cartoons used to refer to "hamburger sandwiches" which were by all accounts the same as burgers today; the "sandwich" part got lost sometime before it got to Australia, i guess.
And fairy bread is just fairy bread, you’re not putting two slices together.
Basically the Aussies picked up burger as from American influence, changed the meaning, then decided that the yanks defined it wrong. Meanwhile all the other bun based sandwiches get to keep the meaning. That said... the international chains haven't helped the matter.
I’m not claiming anyone’s wrong, it’s just what society you grow up in deem it to be.
I am living in Australia though so might be biased here