A 1964 Daily Leader Post article said that the banner was turned over to Summit County Board of Agriculture at the Centennial 4th of July Celebration in 1876. That story does not give a reason why Medina surrendered the flag to Summit County. However, a history of the Medina County Fair, tells a slightly different story.
In 1878, the Medina Fair moved from a smaller site to its present location between Lafayette and Smith Roads in Medina. One of the prizes given out at the fair that year was the Sleigh Banner, which was to go to the county that could bring the largest delegation to the Medina Fair. Summit County won and their Board of Agriculture took home the banner.
http://www.medina-fair.com/general_info/History
Summit County Fair Board, the descendant organization of the Summit County Board of Agriculture, is looking for anyone who knows what happened to the banner. Have you seen it??
A very entertaining account of the sleigh riding competitions of the winter of 1855-56, can be found on pages 75-85 of Those Were The Days by Charles Asa Post. Published in 1935, the main text is about the sleigh races that were routinely held on Euclid Avenue in Cleveland from the 1870s through 1910. But this one chapter is about the earlier competition.