At the NSW BDM online index for deaths, the given names of the father and mother of the deceased are usually listed. I am confident that on both John's and James' d.c.s you will find that their father was recorded as John WESTBURY. One is several decades before Hannah died, and one is several years after her death.
John's 1936 d.c. is #5319, registered Burwood District... parents as John and Hannah
James' 1957 d.c. is #11482, registered Newtown District parents as John and Hannah
And here is Richard's
1976, #22846, (no district displayed as the system was being computerised and all documentation sent directly to Sydney without going through district offices) parents as John and Hannah.
There are several sections of a NSW BDM death registration, and the main reason that the funeral director lodges these forms is simply that the d.c. includes the equivalent of the burial order, so you learn about the deceased and his/her marriages and descendants and his/her parents in one section that the funeral director gets the informant to sign off, and in another you learn the cause/s of death certified by a medico, and in then in the last section is the funeral director's section with the cemetery/crematorium info. The other significant matter to mention is that there has never been a formal cross checking system in the NSW BDM administrative regulations, so these are 'official records' and may contain errors of fact.
(my living elderly rellies include retired senior officers from NSW BDM, and while they are not RChatters, they do read my posts online without signing in)...
JM