"Admission as private patient to public hospital"

I would like to use my private health insurance (which covers me for public hospitals), but have been told by Peel Health Campus that they are a private Hospital and therefore my health insurance will not pay, even though everywhere you look on the internet states that it is both public and private, even the Health Department couldn't understand. So tell me, when a patient wants to use their local hospital and their health insurance (which is meant to take the pressure of the public health system), why would a public / private hospital deem themselves as private when they are going to get paid anyway?

http://www.peelhealthcampus.com.au/About-Us/About-Our-Health-Campus

- "... a 193-bed public and private hospital with 152 acute care beds."

- "... Peel Health Campus is managed by Ramsay Health Care under a service agreement with the State Government of Western Australia. Under this agreement, the hospital treats public and private patients."

http://ww2.health.wa.gov.au/About-us/South-Metropolitan-Health-Service/Our-services/Our-hospital-network

- ".... a 140-bed public hospital and is managed in partnership with private provider Ramsay Health Care Ltd."