I was able to rotate the grooved tube 180 degrees with a flat screwdriver wedged into the groove. I cleaned the area and applied a bit of Crazy Glue around the tube. The shifter cable is now in contact with the unworn part of this tube and excessive friction in the system is gone.
Sounds to me like you've got Campagnolo-sized shifter ends. The difference is subtle but definite.
Left is Shimano, Right is Campagnolo.
Not sure how you're measuring, using a ruler might explain the difference. The common-quoted diameters are
Shimano shifter end barrel diameter: 0.173" (4.39 mm)
Campag shifter end barrel diameter: 0.159" (4.04 ...
This obviously depends on the location on where you ride, whether or not your bike happens to be a fixed gear bike (most are not), whether or not the laws consider the ability to backpedal on a fixed gear bike a brake device.
This could also depend on the specifics of the bicycle. The lawmakers obviously don't want everyone riding on bicycles with only one ...