Up til now Ive used 1 3/16" hole saws for Japanese buttons, and they work perfectly. However, the hole saw may be working it out to slightly larger as it oscillates. My assumption is that with a forstner bit there would be less oscillation as you drive the hole, making a more precise hole.
How well do the Japanese buttons fit using the 1 13/64"? Do they fall in? Do you have to push them in? Would the fall out of the hole if you turned it upside down and gave it a little shake?