Box 19 (of your W-2) is the amount of local income tax withheld. Box 20 identifies the local taxing entity (either a school district or a municipality) for which the tax was withheld.
If there is a 4 digit number in box 20; then it is, almost certainly, the school district.
Ohio employers are only required to withhold for the work city, although many will voluntarily withhold for the resident city or resident school district, as well. You may have to ask your employer who got your local withholding.
If there are two numbers in box 19, then it appears you had withholding for two taxing entities. Box 20 would identify the entities. It should not be two school districts (you do not pay tax to the work location school district, but you do have to pay to the work location city, as well as resident city).
