Yesterday - after 20 working days of utter silence, then a panicked acknowledgement and a very dubious 20 day extension for "consultation" - I finally got a response. According to NZDF,
http://notify.nzdf.mil.nz is the name of an access point from which members of the NZDF as individuals, exercise their choice to use the internet on an occasional and moderate basis. Individuals may visit sites such as yours. Consequently all sites visited by these individuals currently have a referrer of http://notify.nzdf.mil.nz
Which neatly solves that mystery (for the record, I was expecting that it might be media monitoring, as many agencies do). At the same time, I'm left wondering why it took them 40 days to provide such routine information, or why the @NZDefenceForce twitter account, currently proudly advertised on their website, took itself private after finally noticing my request. Is NZDF really giving up on social media just because someone asked them a question using it?
(And yes, you can OIA agencies using Twitter or Facebook. The Ombudsman has guidelines on the subject here)