At least under one test, the threshold for arrest is when a reasonable person would understand himself or herself to be arrested. Although police are generally permitted to lie to suspects about a number of things, I think every court would hold that a reasonable person would think himself to be arrested when a police officer tells him or her that he or she is under arrest. At such point, the arrestee is not free to simply leave, and any attempt to do so, no matter how minute, would be a separate crime (fleeing/eluding, resisting). The upshot of all of this is that, no, police probably cannot lie about whether or not you are under arrest, and any person who would act under such a misimpression would do so on penalty of an additional charge, which could stick even if the base charge is disposed of by nolle prosequi or trial on the base charge results in an acquittal.
Is the requirement to produce identifying information triggered when police demand documentary identification rather than asking for the information listed in Ohio's statute? What about when the detainee attempts to give the required information (the police failure to ask for it notwithstanding), but is interrupted or told that he could be lying?