Q. Do I need to file a return with Indiana?
A. Usually no, but apparently Yes in your case.
" City and state taxes were withheld in Indiana" indicates that you were paid as a regular W-2 employee.
Ohio has a reciprocal agreement, on wages, with all neighboring states. No state taxes are withheld or due and you do not normally need to file a return for the other state (MI, PA, KY, WV, IN). But OH will tax you on your other state income.
If other state taxes are mistakenly withheld, you then have to file a return, for the other state, to get a refund. OH will not give you a credit
However the reciprocity agreement does not apply to local city/county taxes. You can not get those refunded, but if you live in an Ohio city with a tax, they may give you a credit or partial credit. Many do. You need to check with your city.