This doesn't fall into the public land categories, but there is a large
chunk of land (I think in southeast AZ) that is owned by a private group
(Nature Conservancy???).  They allow the land to be used for
recreational use (camping & hiking only I believe).  They restrict the
number of people using the land and do so by a reservation permit
system.  I don't know anything more than this vague information.  I have
a co-worker who did some hiking and camping on their land about 8 years
ago.  Anyway, it's land that you don't want to put a geocache on.
Perhaps someone else knows more.

You didn't include National Parks and National Monuments in your list,
but those rules of use are reasonably well-known and easy to find.

City land (Phoenix Mountain Preserve for example) probably has specific
rules related to it (like don't burn down houses nearby).