Agreed its very fine line that divides to two - for a park where visitors to caravans can freely come and go the the site, would suggest permissive.
Conversly, if a barrier or checkpoint was present excluding all but residents, then I would agree, private is more apt.
From my distant recollection of this site, it was feely accessible, but that may have changed since then. Feel free to change if thats the local mapping protocol.