Hi Kim. I think a species counts as a native if it establishes itself in New Zealand by itself ie without human assistance. The silvereye is a NZ native even though it established here in the mid 1800s from Australia possible because human forest clearance in NZ and maybe cultivation/orchards offered a range of niches that silvereye like that weren't here pre-human when sulvereye probably also blew across the Tasman but didn't establish here.
Where it gets interesting is if after many generations of silvereyes breeding in NZ, the NZ silvereye becomes genetically different from the Australian silvereyes. If this happens it would then be an endemic species ie not found anywhere else.
Both our white heron and white faced herons are native but not endemic species. The pied fantail on the other hand is native and it is an endemic species because it shows distinct genetic variation from the Grey fantail of eastern Australia that looks similar but is now distiinctly different.
It doesn't make the white heron or white faced heron any less special to NZers.
A really puzzling one is the black swan. I had always known that these were introduced by humans to NZ in the 1800s so they are not native. We had a native swan but it became extinct in pre -European times. Recently DNA testing of NZ black swan and the extinct swan I understand has shown that they are identical. In other words when we introduced the black swan from Australia, we were actually re-introducing a native bird that had become extinct in NZ.
I'd be delighted if anyone else can shed any light on this black swan story.