A PIECE OF LAND BEING OWNED, makes it private property.
It may be "public use", but it is still private property, owned by the state.
It's not different that a walmart. It is privately owned property, but intended for public use.
It is not like Wal*Mart. Here are the definitions. Bold emphasis is mine.
Private Pri"vate (?; 48), a. [L. privatus apart from the
state, peculiar to an individual, private, properly p. p. of
privare to bereave, deprive, originally, to separate, fr.
privus single, private, perhaps originally, put forward
(hence, alone, single) and akin to prae before. See Prior,
a., and cf. Deprive, Privy, a.]
1. Belonging to, or concerning, an individual person,
company, or interest
; peculiar to one's self; unconnected
with others; personal; one's own; not public
; not general;
Public Pub"lic, a. [L. publicus, poblicus, fr. populus people:
cf. F. public. See People.]
1. Of or pertaining to the people; belonging to the people;
relating to, or affecting, a nation, state, or community;
-- opposed to private
; as, the public treasury.
Wal*Mart is a corporation, which is privately owned by individual shareholders. It is not owned collectively by the entire public, as a national forest, a local city park or a state university is.
I'm a bit surprised that I needed to define public and private. I had assumed that these are common terms and we were operating from a mutual understanding of the basic definitions, but I guess I should have known better when I read about privately owned public land, or publicly owned private land or whatever mutually exclusive terminology is being used. And these terms are definitely mutually exclusive. Just so we're clear, that means property cannot be both public and private. That's indicated by the definition of private when it says, "not public".
Private land is owned by individuals or corporations. Public land is collectively owned by everyone.
If a piece of land is owned by the state of Kentucky, it is not private property. It is public property. Ownership issues aside, if an institution receives over $300 million of our state taxes a year, I think that had better not be a private enterprise. It had better be a state (public) institution. I don't like it when my Kentucky tax dollars are used to deny me my rights that are enumerated in the Kentucky Constitution.
If Wal*Mart wants to deny me the ability to carry a firearm in their store, it's their property and thus their right to make the rules. As a consumer, I can choose to shop there without my firearm, or not shop there. I can even choose to be lawfully arrested for trespassing for not leaving when the owner of the property or their duly designated representative asked me to leave, although I wouldn't choose to do that. But the state has no right to infringe upon my right to keep and bear arms on state property. Why? Because the property is owned by everyone and that includes me. In a very real way, I'm a part owner of all public land, and so are you.
To my knowledge, universities are the only state properties where the state of Kentucky attempts to enforce anti-gun laws. Presumably, there is some special sentiment related to protecting children, as if 18+ year olds are children. What I'd like to know is whether these misguided sentiments have been codified into law. If so, I'd argue that they are unconstitutional laws.