To the best of my knowledge both the 4162 and the 4137 work with the 124 T ring gear. I had the same issue where about 25% of the time the drive would not seat all the way into the ring gear and when the electrons hit the motor it would just make an awful screech. If you do this enough you could harm the ring gear as Bob mentioned.
I replaced the starter drive, no change. Then I replaced the ring gear, no change. Then I had the motor rebuilt, no change. Again new starter drive, no change. Then screwed the starter button on the motor all the way down so the lever could engage the ring gear as far as physically possible before the electrons hit the motor (should have done this first!). This helped a bit, but I still had the problem.
Finally, yanked the motor out and replaced it with one of the little gear reduction racing motors that you see advertised for the L Head engines. Voile! Success! Now I have a key start and it engages every time without a problem.