Australia an island?

For what it’w worth, Australia’s Department of Foreign Affairs and Trade, on its official website states:

I thought I read somewhere that the determination of whether something was an island or a continent was whether it had it’s own continental shelf. In other words–a continent does, but an island is part of another “continent’s” continental shelf (or are current or former volcanoes). Does that ring a bell for anyone, or did I just make it up?

Straight Dope Column:
Why do Europe and Asia count as two continents?

Straight Dope Staff Reports:

Why is Greenland considered an island but not Eurasia?

Why are Europe and Asia considered separate continents? What’s a continent, anyway?