The continent *is* Australia the landmass - not anything else. And being Australians we were correctly taught ;-p
Oceania describes the general area including Australia, NZ, PNG and the islands in the South Pacific. I think it was the technically correct name for the region we were taught. Australasia is the same area - just a more colloquial reference and I think a bit biased if you don't come from Aus.
I think its a bit like Aus and NZ being referred to as antipodean which is actually a group of islands off NZ - see http://ortho.linz.govt.nz/nz_offshore_islands/antipodes_high_res.jpg