I really don't know, because I don't think there is a definition that would generally agree with both the socio-political boundaries and the geographic/geologic plate boundaries.
For example, I used North America and South America being on different plates as justification for my argument that they are separate continents. However, I can't use that argument for Europe vs. Asia, because they are pretty much one big plate. I would argue though for the separation between them because of the difference in cultures, or possibly a difference in biome or climate.
I really don't care what people say as long as they are precise, and as long as everything has a definition. "Australia" means a certain large island, and "Oceania" is that island plus the surrounding region. "America," from my perspective, means the United States of America. I didn't invent the term, I just grew up with it. "North America" means a specific area between Panama-ish and the North Pole, and "South America" means everything between Panama-ish and the edge of Antarctica.
People in the group I linked to seem to have a problem with people from the USA being called "Americans." I'm not sure why, and I really don't think it matters. In paleontology or biology, we could write a paper and give reasons that the name "America" is misapplied to a certain organism and put forth a new name for that organism. It would still be up to the rest of the world to decide whether or not our reasoning is justified and whether they should follow it.
It's still an interesting question. How many continents do you think there are/should be?