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.