So before Thanksgiving of last year I've installed at home rebuild high mileage starter from the junk yard to see if it will make this kind of noise at cold start:
I installed new aftermarket solenoid form store brand name rebuilt starter (found on the junk yard later) and few other plastic and rubber items.
All metal parts, gears and bushings were not replaced
I've greased a planetary gear set and cleaned the sliding bushing of the pinion gear.
So far there was no grinding sound during cold start, even at 39 F, while last year my original starter made noise at least once a week. If noise returns next winter, the issue is gear set related, if it is going to stay quiet, the solenoid was root cause.
Added later: the "dry gear" screech noise common on VW Jetta is possible due to the metal ring gear in the planetary set

note the steel ring gear on Jetta starter

Fit starter has plastic ring gear, so most likely the noise recorded in video is made by a faulty (sticking) solenoid plunger