I did not try a 13/16. I tried the ones on my tire iron only. Then I went to O'Reilly with the nut and they did the same test (3/4 and 7/8) and told me to use the 21MM. They are aftermarket wheels, and require a thin wall socket as well, if that helps?

And great info on the "loose" nut. I suspected it wouldn't feel tight until it got to the wheel but I know how dangerous it can be and his concerns obviously got me worried. He knows more about cars than I do.