I think they always call it a “movement check”, but never give more details. I bet they use the same term from simple rate adjustments to full on movement overhauls. :/ Richemont group brands are much better in terms of describing the work done imho, but that’s a discussion for another day

Are you able to re-create the problem of the watch stopping again? At least from my last training, Rolex uses a 4-day timing test, where they check the watch in static positions and also on the winder to test the rate, amplitude, and power reserve. If the watch stops or doesn’t meet certain tolerances, the test helps identify what could be wrong.
If you have access to an AD with an on-site watchmaker, I’d see if they could do this test, or at least put it on a timing machine to check performance.
Rolex themselves do not have an exchange policy for problematic watches, but WoS might. It’d be worth reaching out to them too.
In my experience, it’s quite rare to see a Rolex with performance problems so early (they have one of the lowest warranty repair rates in the industry). And their service center repairs are very good. If you do send it back to Rolex, give the service center number a call; they may be able to run extra checks for you. Good luck!