I have lived in Ohio for almost 20 years now and I’ll start this by saying it doesn’t matter with regards to the middle name vs middle initial thing. My drivers license says “Colin K Taranovich”. To renew my car’s plates, which on the registration only shows my first/last name (no middle name at all), they gave no trouble at all. Same thing with my Ninja, no trouble at all, even though it’s registered in a totally different county where I no longer live (my parents house).
Point is, every document from the BMV I have ever had shows my name multiple ways. Full name with full middle name, full name with middle initial, or full name only (no middle name or initial at all). Never once have they hassled me over it even though like I said, my drivers license shows a totally different address than my car and motorcycle registrations.
The important thing to take from this, is that Ohio requires you to bring a primary ID, and a secondary ID. Primary is drivers license, passport, or birth certificate. Secondary ID, you have a bunch of options. Just to name a few, SS card, college student ID, concealed handgun license, utility bill, military discharge paperwork, W2s and other tax documents. List goes on and on.
Full list is posted on their site. Just gather up literally every one of the ID forms that you have on that list, and bring every single one with you. Shouldn’t have any issues.
EDIT: Here’s the official list of what they accept:
http://publicsafety.ohio.gov/links/bmv2424.pdf