When Australia became a dominion in 1901, it chose to call itself the "Commonwealth of Australia", but when Canada became a dominion in 1867, it was called the Dominion of Canada and is now just "Canada". The term commonwealth is pretty old in English law, why wouldn't Canada use it?
What a coincidence, today I was interested in finding out when Canada became fully independent from the UK even from the Statute of Westminster (1982, but that's a story for another day) and I cam across almost this exact piece of information!
The Dominion of Canada was originally to be called the 'kingdom of Canada' but the British Foreign Office warned that the USA would not take kindly to a kingdom on their border. According to sources Paslm 72:8 inspired Leonard Tilley one of the I believe Canadian 'fathers of the confederation' to call it a dominion instead. The psalm read '[Jesus Christ] shall have dominion also from sea to sea, and from the river unto the ends of the earth'. The 'from sea to sea' would later be rendered in latin' on the seal as 'A mari usque ad mare'
Hope this helps, I'm new to reddit and still getting the hang of it!