In Ohio, it is illegal to drive any motor vehicle without insurance or other financial responsibility (FR) proof. It is also illegal for any motor vehicle owner to allow anyone else to drive the owner's vehicle with FR proof. When applying for a driver's license or renewal of such license, Ohio applicants sign a statement indicating that they have proof of FR and will continue to maintain FR or will not operate a motor vehicle in Ohio. This is why even if you do not currently own a car; you still need auto insurance in OH.
Non-owner insurance is car insurance that one can purchase if they do not own a vehicle of their own but will be driving other vehicles and want to have the liability insurance required. Non-owners insurance does not typically provide collision or comprehensive coverage on the vehicle, so perhaps that is what made you believe this type of insurance had no coverage for the automobile.
Now bond insurance is sometimes another term for a SR-22. In Ohio, failing to provide financial responsibility results in violating the FR law and being required to file with the Ohio Bureau of Motor Vehicles (BMV) a SR22 or bond, thus continuously maintaining proof of FR for 3 to 5 years.
If you are required to obtain a SR-22 and do not own a vehicle you can acquire a non-owner SR-22. With this insurance, you will fulfill the requirement set forth by the BMV and have liability insurance coverage while driving vehicles, outside of your household, owned by other people.
If you need a non-owners or SR-22 non-owners policy, you can begin the quoting process for Ohio auto insurance online here.
Comment Update: Not all insurance companies offer all types of coverage. For example, you will have to find an insurance company that offers non-owner insurance and/or an SR-22. I know it sometimes feels like the "run around" when it comes to compying with the law, that is why we offer this service. We may or may not have a company in Ohio that will offer non-owners insurance.