Grey13, I will try to explain..................
CL sites are allowed to operate because they have an 'exemption certificate' issued by the Caravan Club (CAMC).
This allows them to have a maximum of 5 caravans on site at any one time, subject to other conditions, and specifically a maximum stay for any caravan of 28 days. This avoids the need to apply for planning permission in the usual way. The Club also specifies certain requirements regarding fresh water; toilet disposal, etc.
CS sites operate under a very similar arrangement operated by the Camping & Caravaning Club (C&CC). A significant difference is that in addition to 5 'vans, the site can also accommodate tents.
They are all completely independent privately owned and operated businesses, and may set whatever conditions they wish. In both cases you have to be a member of the relevant Club in order to be able to use them.
(Just for completeness, there are a very few other '5-van' sites which operate under exemption certificates issued by other bodies).