I think in the retail world the term refers more to being identified down to name and address so a letter of notification can be sent telling you that you are forbidden to come into the store and that the next time you come in the store you will be arrested for trespass.
I get the "don't come back" part. But what no one has answered is the
first instance question. Can you be trespassed without your knowledge?
Example:
The store manager says "
get out," but he
does not say stay out on the first instance. You go back in the store the next day because the manager did not
unequivocally say "stay out." The manager must say to you, if he knows you are in the store again, "get out
AND then stay out." The letter
is not a confirmation of the first instance of being asked to get out but a confirmation of the first instance of a stay out order.
Record record record.
Though, that likely will not matter to a cop. Having "evidence" that you were not told to stay out can/will only be proven to a judge.