You’re an archipelago

There is no agreed upon definition or set of continents globally

It might as well be a made up word

The Australia thing is something we were taught in high school.

I may be mistaken here, but I think the concept of a "continental shelf" is pretty well defined geologically. That is: Outside a land mass, the ocean floor extends a certain distance before dropping off to the deep ocean floor. An island would be a piece of land that sticks out of the sea from this continental shelf, while the "continent" includes the entire shelf, and all the land masses that stick out of the ocean on that shelf.

Of course, this seems to break down a bit for e.g. the Europe/Asia divide (and probably a lot more), but the concept of "continents" vs. "islands" can make sense geologically, although the "continents" are then different from the geopolitical borders ones we usually talk about.

By definition, a continent and an island are mutually exclusive. Nothing can be both.

Australia isn't a continent either. It's in the continent of Oceania, which includes New Zealand.

Antarctica is a continent in its own right. It's not a country and not a nation.

Not all continents, no. Islands are bodies of land surrounded by water on all sides, no? Wouldn’t Antarctica and Australia then qualify by that definition?