If the criteria to be public housing is that someone manages the lifts, lights, and exterior. Then all apartment blocks are public housing. Condos and HDBs are all public housing.
Your logic is infallible. Of course, your premise is rubbish.
It's not about whether 'someone' is managing these things. Even an owner-occupier is managing their own lights and exteriors.
A useful criterion is whether the government (or an arm of the government) is managing these things. Like HDB, but unlike your typical condo management organisation. It's far from the only criterion, but it's a useful indicator.
"Singapore has one of the highest home ownership rate in the world, due to the commitment of our founding fathers since the very beginning of nation building. Today, there are more than 1 million HDB flats across the island, home to close to 80% of Singapore's resident population. The schemes highlighted below help Singaporeans meet their housing needs at different life stages."
So the dictionary says that HBD housing is public housing, wikipedia says that HBD housing is public housing, and the Singaporean government says that HBD housing is public housing.
It's not about whether 'someone' is managing these things. Even an owner-occupier is managing their own lights and exteriors.
A useful criterion is whether the government (or an arm of the government) is managing these things. Like HDB, but unlike your typical condo management organisation. It's far from the only criterion, but it's a useful indicator.