What is the legal definition of a common -law wife?

So what is the significance of the “common law” part of the term common law marriage?

States have statutes providing for valid marriages. The common law marriage does not comply with the state statutes. A statute in derogation of common law must be strictly construed. If the statute does not abolish explicitly the c/l marriage that remains a viable alternative. Prior to its abolishment in England in the 18th C the c/l marriage was a valid marriage

In Australia if you live as a couple you are in a de facto relationship which is essentially the same as marriage. Can be same sex as well.

Lord Hardwicke’s Act was in force in Upper Canada, via the principle of statute reception.

Yes. It didn’t directly apply in the colonies as an Imperial act overriding existing colonial law. Whether it applied in any particular colony depended on whether it had been incorporated into the law of the colony by reception, adoption or reenactment.