Owned by a state agency, either directly or indirectly.
So yes, Newcastle is owned by PIF, which is a state-controlled sovereign wealth fund and therefore imho Newcastle are state-owned.
City are owned by an individual. Yes, I know he is a very important individual, is closely related to the Emir and lines between authority, influence and control can be blurred sometimes but, legally afaik, the club is not state-owned. I had wondered if the change in Mansour's choice of holding company (from ADUG to Newton) was something to do with a PL challenge on that, but it has never been mentioned anywhere.