I don't have a Cox tractor, so I have to work from the parts list. A common way to engage and disengage cutters on ride-on mowers is to have a pivoted idler pulley running on the cutter belt. When you put the lever in the engaged position (forward) the linkage pulls the idler against the belt, tightening the belt so the cutter runs. When you pull the lever back, the idler is pulled away from the belt, leaving it slack so the cutter does not run. I suggest you disconnect the engine's spark plug, then have a look at the cutter belt while you move the cutter disengagement lever back and forward.
(Never look under the tractor without disconnecting the spark plug.)
