The information you have found from the Immigration website is referring to "NZ Aid Student Visa". This is a special category of student visa. If your partner holds a Student visa as part of the NZ Aid programme (in other words a government scholarship student), then you can apply under this policy. If the student visa is a normal student visa then you can only have a partnership work visa if your partner one of the following:
1. people granted student visas to study for a level 7 or higher qualifications in an area of absolute skill shortage as specified in the Long Term Skill Shortage List; or
2. people granted student visas to study towards postgraduate qualifications;
If you are unsure of whether your partners Diploma programme makes you eligible then you can mention it here.
If the Diploma does not meet the requirements then you can only get a visitor visa as the partner.