You don't need to notify INZ - it's none of their concern any more.
When anyone applies for residence because of partnership, they check that the couple really have been living together for at least 12 months, and still are. Once they know that, and issue the residence visa, that residence does not depend on anything any longer. You are a resident, just the same as someone who got residence because of a skilled job. It's YOUR residence, free and clear.
The next thing you need to keep in mind is that in two year's time, you can apply for Permanent Residence (which allows you to live and work in NZ for ever, and also come and go whenever you like with no restrictions), but you need to qualify for it - check the 'Evidence' links down the side here.
https://www.immigration.govt.nz/new-...ZL&applying=no