Special Category Visa Sub Class 444 allows NZdrs to enter Australia indefinitely but see this:

The Special Category visa (SCV) is a temporary visa electronically granted to New Zealand citizens who arrive in Australia on a New Zealand passport that is in force and who do not hold a visa.

You must:

• be a New Zealand citizen
• meet health and character requirements
• not have been deported, excluded or removed from any country.

Note: If you do not meet these requirements, you may not be eligible for a SCV, and you may be refused entry to Australia. Please approach the nearest Australian immigration office or visit www.immi.gov.au for further information regarding entry requirements for New Zealanders before travelling to Australia.

The SCV allows New Zealand citizens to lawfully live and work in Australia indefinitely. The SCV is a temporary visa and, except for certain holders resident in Australia on or before 26 February 2001, does not provide a direct pathway to citizenship. If you wish to become a permanent resident to qualify for Australian citizenship see below 'Pathways to permanent residency'.

Further on the Dept Of Immigration says:

In certain circumstances if you were a SCV holder on or before 26 February 2001 you may already be considered a permanent resident for the purpose of becoming an Australian citizen, sponsoring a family member for permanent residence or accessing social security payments. If this is not the case, you must apply for and be granted a permanent residence visa. The various categories of permanent residence visas are summarised in the Migrating to Australia information form. For more information about Australian citizenship for New Zealand citizens contact the department on 131 880.

So the DDCA might just be wrong about NZ citizens being Automatic Permanent Residents.