The applicable Chauffeur-drive terms & conditions are those of the carrier operating the flight for which the service is used to get to or from the airport. This applies regardless of whether your ticket is a codeshare ticket issued by a different carrier.
Itineraries that are exclusively between New Zealand and Australia on Emirates-operated flights are not eligible for Chauffeur-drive. All other itineraries on Emirates-operated flights are eligible, including Business Class itineraries that have a stopover in Australia.