Just because land is publicly owned does not make it public space (No 10, Downing Street, for example) and apparently public spaces can be privately owned. There may be regulations in place to allow access (as in the Sky Garden at 20 Fenchurch Street in London, concluded as part of a s106 planning agreement) or owners may welcome visitors (as in shopping centres). Even without the consent of the landowner, there may be footpaths across land or rural and coastal areas may be mapped as "access land", enabling right to roam. There is no specific legal designation for public space and it is the attitude of the landowner - whether public or private - that is the prime determinant of what is permitted in (apparently) public space.
I have written two papers on public space from a property point of view to explain the legal context. One long (here) and one short (here).