Why does a new starter drive continue to hang up on cold mornings?

There is no "summer" or "winter" lube for starters. Sounds like the old flubber gasket that service station people would tell a woman their car needed. Have you inspected all the teeth on the flywheel? Obviously, if you have the correct starter ( I would get a 2nd opinion on that) the starter is NOT the problem. If it is a manual, have you had any previous clutch or pressure plate work done? You pretty likely have some clearance problem that is being caused by another part of the starting assembly IF you are truly getting the correct starter. Again, I'd try another shop or parts house to make sure you are getting the correct starter. If you are, start looking at the teeth. A possible very long longshot is that the starter is not getting enough current to function properly. A rare occurrence but check the power at the starter when activated and make sure it is the proper voltage. Insufficient power due to a faulty wire or low battery may not be enough to fully retract the bendix. This could be why its happening only on cold days when the starting load on a battery is highest due to cold oil and if the battery has set overnight. Have a free 'LOAD' test done at you parts store to make sure you battery is up to snuff. Also make sure your battery has the correct CCA (cold cranking amp] rating for your car. Also check the starter wires for continuity.