I hate internet shorthand. I think what you are describing is a diversion program. Most diversion programs automatically expunge (seal) the record. Go the clerk of courts website and see if their case records are available online. If you can still pull up your case (and the case occurred within the last fifteen years), then it means your case wasn't expunged. If so, or if the court doesn't generally have case information from the time of your case available online, AND YOU HAVE NO PRIOR CRIMINAL CHARGES (not including traffic tickets like speeding) or any pending, then you can petition the Court to expunge and seal the court's record as well as the arresting agencies' police reports.
An expungement merely removes the case from the public record. Ohio statutory law does provide that information about the existence of your case can still be disclosed and discoverable under limited circumstances (certain state licensing authorities, for example, may be able to review expunged records.)
Nonetheless, there is value in having an expungement as it removes the incident from most criminal background checks.
Consult with a criminal defense attorney in the area where your case occurred and they should be able to file for your expungement for a modest flat fee.