Is New Zealand considered part of the continent of Australia?
It does not say that. It says “New Zealand is Not part of the continent of Australia…. but is part of a submerged continent called Zealandia”. This is a ridiculous statement. Submerged continent is a meaningless term that could include anything you want to include. New Zealand is a group of islands, Australia is a continent. Sometimes, when islands are part of the same continental plate as a continent, they are considered part of the continent. Or when islands are part of a country on the continent they may be considered part of the continent. New Zealand is neither of these. It is geologically and politically separate from Australia.
