Contrary to the point made somewhere here about how "Oceania is an outmoded espression", it is actually the newer concept: The word "Australia" has meant something about that part of the world for longer than has "Oceania". (At a guess, the Olympic rings symbol was designed and acquired the description of what it stands for before "Oceania" came into widespread use.)
The whole probelm here is that the English word "continent" actually means two different but closely related things, namely the older cultural-geographical concept "part of the world" (cf Ger. "Erdteil", lit. "part of the Earth") on the one hand, and the more narrowly geological idea of continental shelves &c. "Hello, so which part of the world are you from?" is a much older question than the idea of tectonic plates, so Europe and Asia being different things has been thoroughly cemented into the public conciousness although geologically, they're one and the same. Contrariwise, the Americas both got the same name even though they're (apparently, AFAIK) actually two different things. So, it would help immensely to keep these two different categorisations separate. "Erdteil" is often called "region" in English, but that word is also already overloaded with quite enough meanings, so this usage probably doesn't exactly bring any increased clarity on the whole.
Unfortunately, contemporary colloquial English uses the word "continent" for both concepts, so confusion like yours is only all too common. To come back to your specific points:
* The ring represents Oceania and not Australia
In modern politically correct parlance it represents a part of the world ("Erdteil"), not a single geological entity. Saying it represents only the actual continent of Australia would leave New Zealanders and Micronesians without any representation at all, so that would be rather un-PC. My guess is that back when the rings symbol was adopted, PC details like this just weren't on the agenda, so they simply said "Australia".
* and this makes Australia an island in Oceania.
No it doesn't. It makes Australia a geological-continent in the region-of-the-world Oceania; or, if you will: It makes Australia a continent-1 in continent-2 Oceania. (See how using the same word for different things kinda sucks?)
(One odd example is the five Olympic rings, which represent the "five inhabited continents": Asia, Europe, Africa, Australia, and the Americas.)