Hi James, great to see you having another crack at some of the more unusually names NZ regions (including my home city and region of Auckland…)
The regions listed above are correct, though I have seen occasionally Tasman-Nelson listed as one area.
I’m not sure how aligned to OSM data you need to be - I’m guessing very - and can see why you have issues with NZ based on how things are mapped there. For Auckland, you could look to use the Administrative levels as the next level down. These are at admin level 8 but there doesn’t look to be an admin level 6 for Auckland (which would be what we would call Auckland City and encompasses most but not all of the level 8 administrative levels).
Not sure how to treat Nelson but the level 4 Region is small, there wouldn’t be much outside of the city.
For BoP/Rotorua, not sure how else you would do it and I think based on what I see looking through Nominatim is the only way it could be added (and this matches what I can see for the likes of Wellington City and Christchurch City)