While that definition might not hold water for some people, there is no official globally recognized definition of a continent. Just as some seas are called seas and others are called gulfs or bays, continents generally refer to the major land masses of earth.
Even though Australia is the smallest of the accepted continents, Australia is still more than 3.5 times larger than Greenland. There has to be a line in the sand between small continent and the world's largest island and traditionally that line exists between Australia and Greenland.
Besides size and tradition, one can make the argument geologically. Geologically, Australia lies on its own major tectonic plate while Greenland is part of the North American plate.
Locally, residents of Greenland consider themselves islanders while many in Australia see their county as a continent. Even though the world lacks official definitions a continent, it should be concluded that Australia is a continent and Greenland is an island.
On a related note, I will here state my objection to including Australia as part of a "continent" of Oceania. Continents are land masses, not regions. It is completely appropriate to divide the planet into regions (and, in fact, this is quite preferable to dividing the world into continents), regions make better sense than continents and they can be standardized.