I've always been led to believe that OTW meant Out To Win, in that
Meyers was "out to win" a primary trainer contract with the military. I
haven't heard the OTK monicker, maybe some folks use that to distinguish
between the Warners and Kinner.
OTW-125, 125hp Warner Scarab
OTW-145, 145hp Warner Super Scarab
OTW-160, 160hp Kinner R-56
Almost got a ride in an OTW a few years back, but when my turn came
we found the tailwheel had let-loose of it's ball bearings...had to settle
for a Stearman ride instead.

Edit: Which is a bit ironic, considering
that Stearman got the contract...