By Law, If a private business tells you to never comeback, they probably will follow up with a Criminal Trespass Affidavit. That is a crime if you return. It was taught in the Academy to have a few locations pre written for Criminal Trespass and Larceny reports because of the amount of calls you have from those locations.
Disagree. If you are a retired LEO and packing they can ask you to leave. Assuming you comply, they have NO grounds to detain you or ask your name. They could call 911 to report a man with a gun, but when stopped the dispo is PROPER ID. There is simply no way they get to issue you a trespass, however, if they see you there again they could indeed have you arrested.
The general principle is that in a place open to the public such as a store or mall, there is a presumed license to be on the premises.
Once they revoke that license, you can not return or you are subject to arrest.
The things you speak of are after a person is caught stealing. That gives them a right to detain and identify you, and AFAIK if you do not cooperate, they call the PD and have you arrested for petit larceny. So most people cooperate and sign the trespass.
But this is different. You have broken no laws.