Yes, a Spyder is a motorcycle in Ohio and you need either a two wheel or three wheel endorsement to ride one. The written test that you take at the BMV to get your permit is the same for either endorsement. If you then take the skills test (riding test) on a two wheel machine, you get the two wheel endorsement, which is good for either two or three wheels. Take the test on your Spyder, and you get the three wheel endorsement - three wheeled machines only. As of 2016 a vehicle such as a Slingshot is no longer considered a motorcycle in Ohio.
If you want to and are able, a highly recommended way to do this is to take the Ohio motorcycle riders course. You need to have a valid permit and they supply everything else, including a two wheel motorcycle. Pass the skills test at the end of that course, and no further testing is required at the BMV. Just take the completion card to the BMV and they issue you your new license with the two wheel endorsement on it. At this time, you cannot use your Spyder for this course. Some folks like the flexibility of getting their two wheel endorsement up front, then they have it for whatever comes along in the future. The course also provides excellent education for safely riding in this environment no matter how many wheels you have.
http://www.motorcycle.ohio.gov/basic_rider.stm