So does being in the Commonwealth alone make you a monarchy, or is it some extra decree?
In the case of Australia, our Constitution first and foremost makes us a monarchy: The legislative power of the Commonwealth shall be vested in a Federal Parliament, which shall consist of the Queen, a Senate, and a House of Representatives, and which is hereinafter called The Parliament, or The Parliament of the Commonwealth.
(Commonwealth in this case refers to the Commonwealth of Australia - the British Commonwealth/Commonwealth of Nations did not exist in 1900 when our Constitution was written and so there was no need to elaborate).
From 1953, onwards, the Queen within Australia has been known as the Queen of Australia (until 1973, "Queen of Australia, the United Kingdom, and Her Other Realms and Territories", since then as "Queen of Australia and Her other Realms and Territories, Head of the Commonwealth".