It is private property, it is not a public park. The owners can ask you to leave anytime or bar you. Read some of the relevant acts.
They need nothing in writing, you can be asked to leave anytime, this is even true of govt owned property. They can also ask you to stop taking electricity, or leave the restroom. You are their as a guest if you like, they make the rules.
You are making assumptions on what op wrote and claim, but again they can enter for a number of reasons, it does not become "your castle", there are times when they must enter too.
They do not need a sign. You keep coming at this as the person has more rights than the owners, bizzare. It is theft, but I doubt it would ever be escalated to charges, nor would they bother with any sort of other action (the people whom have to pay for the electricity taken)
As to your last paragraph, I get the feeling that the OP knows what they were doing on various levels is wrong, but takes the victim approach and points the finger everywhere but in the mirror.