He literally made the choice to renounce his Australian citizenship. In my mind, that makes him less Australian than some random person who happens to have been born non-Australian and stayed that way.
You are correct and he did do this - but it was purely a move to allow him to own broadcast rights in the USA. He needed US citizen to do that, and it didn't really affect him in any other way, so he did it. He still maintained a base in Australia (as well as UK and USA) for the rest of his life.