Been around since 1981. Not all school district have the income tax. Click on link below, enter your address, and choose 2005 year. Example result from my address below. No rate therefore no school district income tax for me.
School District Income Tax
Year: 2005
School District Name: WESTERVILLE CSD
School District Number: 2514
Income Tax Rate: 0.00%
Department of Education Number: 045047
Note: Do not use the Department of Education's school district number when filing your Ohio income tax or employer withholding tax returns.
https://thefinder.tax.ohio.gov/StreamlineSalesTaxWeb/AddressLookup/LookupByAddress.aspx?taxType=SchoolDist
If it does come up with a rate, you owe school district income tax. If the tax was not withheld, let your employer know what school district you live in, so the tax will be withheld in the future.