Yes, it does seem that way Cando.
There's many a flaw in NSW BDM online indexes for births, deaths and marriages in the years 1856-1890s, so if Phoebe was born in NSW, it is possible her birth has not been included in the registrations listed by NSW BDM.
I am not familiar with NZ BDM for any possible flaws, so I don't know the admin setup behind their central register. I do know though that it was not compulsory for Maori births to be registered until about 1913.
http://library.huttcity.govt.nz/docs/OPAC/PDFS/RG-BDM.pdfAdd
https://www.bdmhistoricalrecords.dia.govt.nz/timelineJM