I doubt you will find a definitive answer to this question, as Ohio did not even require vehicles to be registered until 1905, so there would be no way to know definitively how many operating motor vehicles there were in 1895.
However, if anything, I would suspect that there were no operating motor vehicles in Ohio in 1895.
Wikipedia cites a study by L. Scott Bailey which includes extensive research into early motor vehicles. What is likely to be the first American-made gasoline powered vehicle was being driven around Ohio in 1891, but there was only one; no one else wanted to buy one. By 1895, the car was no longer being used on the road.
Lambert was convinced that he had a workable engine to power his three-wheel carriage and he set forth specifications and a price of $550 in a sales brochure which was mailed during the first part of February of 1891. Later that month, the automobile was running with the new stirrup-type steering on the main street of Ohio City. ... Although the letters of inquiry continued, no sales contracts were signed for the Lambert. Lambert soon realized that there was no sales potential for his automobile.
(Originally quoted from Bailey, L. Scott, "Historic Discovery: 1891 Lambert, New Claim for America's First Car", Antique Automobile magazine, Vol. 24, No. 5, Oct–Nov 1960, p. 343).
The first sales of commercially available cars were made by the Duryeas Brothers, in 1896. As such, it seems unlikely that one, let alone two, people owned motor vehicles in Ohio in 1895.
According to the Ohio History Central web site, Lambert's vehicle was involved in an automobile accident, in Ohio City, which they claim is the first recorded auto accident in history. However, the vehicle collided with a tree, not another car, and happened in 1891, when Lambert was still driving around Ohio City trying to sell his prototype. It's possible that the claim in question is a misrepresentation of this earlier accident.
