Approximately zero of the indigenous population present prior to settlement by Europeans would have identified as being Australian, or of being born in Australia, or anything like that.
Definitively, yes, Australia "began" with European settlement of the continent.
Except I do live here, and it's patently ridiculous to say The Commonwealth of Australia is anything but an invention of European settlers. Of which Queen Elizabeth II was the Head of State for 57% of its existence. Which is the context of this comment thread.
Can't the term "Australia" just refer to the place on the globe? Like if someone was telling me what Antarctica looked like 50,000 years ago, I wouldn't tell them that Antarctica didn't exist back then. I'd know what they meant.
"Australia's history" was a poor choice of phrase exactly because of that ambiguity. In fact it was explicitly referring to the period since 1901, which very few people would consider to be the start of "Australia's history".