Im sure there is some exception, but ive seen the tresspass process by watching a lot of lawyer videos that analyze police interactions (a fun past time of mine) and I think i have a decent understanding.
Simply saying someone is trespassed isnt enough. First they must tell you to leave, if you leave then you cant be id'ed or get the actual official tress pass warning. If the cops get there and they witness you being asked to leave you must be given the chance to leave, if you do they cant ID you or issue the tresspass warning, but should you come back at any time they will issue the tress pass warning, it is not criminal to come back though.
Once you have the tresspass warning then if yous tep ont he property again you go to jail and face criminal charges.
Also a cop can not trespass you, the owner must initiate it. A cop also can not solicit a trespass (that is request of the owner to trespass someone).
That said in some states you can preemtively trespass someone, other states you cant.