Except she can return because she needs to finish her part of the trespass complaint. In short, if you can just leave the scene of a trespass complaint, without giving ID, the criminal trespass law is effectively null and void.
The law in Ohio and most everywhere else is that trespass does not occur if you have been asked to leave and you leave.
Trespass only occurs if you’ve been asked to leave and then you leave only to return.
In this case the woman was complying with the request to leave when the officer arrived. Meaning that under the Ohio law (which I cited in an earlier post) no crime had occurred. The officer then did not have a justification for demanding ID.
Where his own department has put him on unpaid leave for this situation I’d say that his department is reading the law correctly.