Both seems to be the case, to limited degrees. The Harbour Trust website that you refer to I believe does mention that it's now leased as a private residence but is still managed by the trust and opened periodically for public tours:

"The Marine Biological Station today

Today, the Marine Biological Station sits within a resplendent garden of native vegetation and distinctive coral trees. It’s now leased as private residence, but is managed by the Harbour Trust, who periodically open this fascinating landmark for public tours." (https://www.harbourtrust.gov.au/discover-and-learn/harbour-history/historical-places/former-marine-biological-station/)

Does this resolve the note?