India is not a commonwealth, but is a member of the Commonwealth of Nations. Charles III is the head of the Commonwealth but not the monarch, he is separately the monarch of some (but not all) Commonwealth countries. Most Commonwealth countries are republics.
Australia is both a commonwealth (in name) and a member of the Commonwealth of Nations, but those are two separate kinds of commonwealth.
To summarise: All commonwealths are commonwealths but not all commwealths are the Commonwealth of Nations and not all members of the Commonwealth of Nations are commonwealths (although some are).
India is not a commonwealth, but is a member of the Commonwealth of Nations. Charles III is the head of the Commonwealth but not the monarch, he is separately the monarch of some (but not all) Commonwealth countries. Most Commonwealth countries are republics.
Australia is both a commonwealth (in name) and a member of the Commonwealth of Nations, but those are two separate kinds of commonwealth.
To summarise: All commonwealths are commonwealths but not all commwealths are the Commonwealth of Nations and not all members of the Commonwealth of Nations are commonwealths (although some are).