I think people are arguing because they have a different understanding of exactly what some of the words used mean.
By park I mean a big green area which might have trees or a playground inside it, or even a pond and a cafe. It might be fenced off or it might not. It might be inside a council estate or anywhere else in a town or village.
A council estate would usually be roads of houses, maybe with blocks of flats in it. There might also be facilities like a parade of shops, churches, schools or even a big shopping centre all within the council estate. There might be buses going through it. Sometimes the blocks of flats might be inside a big field with a playground. Sometimes the blocks might be close together and the playground close to them. But with no fences.
Private estates would just be a load of houses built by the same developer at the same time and that look the same. They'd usually have roads, shops, schools, parks and playgrounds in as well. Unless they were a smaller and usually more recently built gated estate.
Social housing might just be a private block with its own common garden, not open to everyone. Or it might be mixed in with the council estate.
A playground would be a surfaced area with play equipment on it, or a school playground, which might also be a yard. If the park has no fence or playground it might just be called a field. The park / playground / field would all be for public use.
I live outside London.
I have lived in London though, and I noticed there, that the estates often have more of a defined edge than they do where I live. The playground might be behind or surrounded by blocks of flats on all or most sides. There might be a fence and gate that look more private. There might not be big green areas around it and in between it.
Some people must be thinking more of that kind of set up.