NEVER — not in my PADI training courses to the rescue level and not on ANY dive boat — has the precise role of a DM been explained to me. Nor have I ever seen a scuba operator post this information on the company website.
So, the industry expects divers to know what they've never been told.
I'm going to beg to differ. Or rather, I'm going to claim the "role of the DM" is not relevant to the decisions an individual diver has to make to keep themselves within the standards of safe diving practices.
The PADI OW course explicitly says two things that all students of the course have to acknowledge, with their signature, before being certified:
1. Every diver is responsible for their own safety (and this is an explicit question on the OW exam.)
2. Every diver is responsible for understanding, and staying within, their limitations.
The role of a DM in Cozumel is
(a) point out pretty fishes
(b) hold a buoy at the end of the dive to signal the boat where the DM ended up.
Note that point (b) does not include "the rest of the group". Many times the rest of the group is with the DM, but I've personally had plenty of times that my buddy and I have separated from the group (hanging back to take video and pics, and currents being what they are) and I've had to use my own SMB. If you expect any more than (a) or (b) from your DM, you're violating points 1 and 2 that you promised you would follow when you were certified.
Diving in Coz without carrying your own SMB and reel has a lot in common with Russian Roulette.