Lena Gertrude GAEDNER buried: 27 Sept 1905, Karori
I am not seeing this on the cemetery website. Possibly it is an error to the database.
Use the feedback form on the website, giving name and burial date for Lena, and ask them to locate the burial plot. You need to ask who else is buried in the plot.
Also ask for all burials in the plot of George and Annie. Lena might be buried with them, but if she is not, then it will also be useful to see other names with Lena.
You have given information for three death records. Unless you say otherwise, and because I have asked you, I will assume that you have no other birth, death and marriage printouts. You have no baptism records.
You need this printout -
NZ BDM marriage
1875/849 Annie REID m. William COOK
Marriage records are really important because the bride and groom give information about themselves. Death records show information given by someone else.
I have looked again at the probate file for Annie COOK, and it identifies a plot of land in Wellington, only. I think the land at Timaru might be on file for George Potter GARDNER.
Annie made out her Will August 1895, six months before she died. If Annie died suddenly after a short illness, you might wonder what caused her to write a Will when she did.
If Annie died of a chronic condition, it would fit with her making a Will, and would also cause you to wonder who was caring for her young children during her illness, until her death.