I wonder about your wording .... I have had a good look at various NSW BDM records over some 50 years of involvement in my hobby, family history. I have not ever noticed any dc with the words 'de facto' for the relationship. I would expect that Annie's dc will show the informant as her daughter, and if so then I expect the daughter will be of the view that Annie was twice married.
In the NSW BDM administrative system, there are no BDM officers employed to validate the information given by the informant to the funeral director. The informant signs a section of the funeral director's papers. It is the equivalent of a Statutory Declaration, and there are heavy penalties for prejuring yourself. So the informant provides information that they believe to be true. That's why it is often said that the information on a death certificate is not as reliable as the information on a marriage certificate.
Cheers, JM