There is abit of confusion regarding what a DiveMaster is responsible for.
I will present to you some information so you can not be afraid to dive in Cozumel. You will prepare better because you will realize that their are limitations to the responsibility of the DiveOp and the DiveMaster. And you will be careful to dive as if you were diving independently, because that is what you are certified to do, in the conditions and depths to which you were trained.
In the traditional form, a DiveMaster is usually on the boat, not in the water.
The DM is in charge of doing a weather check and site survey, arranging the boat and equipment, briefing divers once at the site, making sure divers get properly off and most importantly, back on the boat, and to organize a rescue if that is unfortunately required. The boat Captain is master of the ship and not responsible for DM'ing. Many boats are now bareback charters and you can compare the boat to be water taxiis with no other responsibility.
But it has become common in resort areas to have an in-water dive guide to have a local experienced divers in the water, for obvious practical reasons, to show visitors the way, to head off problems, and to point out areas of interest and allow the dive to be that much more enjoyable = happy divers.
However, there is no write up on what a dive guide's responsibilities are. They certainly are not responsible for managing each individual diver's dive.
The DM has a duty of care to the divers, but it is not what you may think, because you may have only been exposed to dive guides. Watch out, you are still responsible for being a thinking, conservative, responsible diver!
The DM's main responsibility, quoting the PADI DiveMaster manual Page 17 is to "generate a broad dive plan that individual divers plan their own dives within, to prepare for reasonably foreseeable emergencies, and to orient divers to the site". The emphasis is that they do not take the place of the buddy system for certified divers.
NAUI Leadership text Page 114 adds "Given the range of diving variables, not the least of which is the divers themselves", dive guides cannot assume the role of personal safety guardian or lifeguard".
Now, back to coppermaus' story. The Devil's Throat is an advanced dive site. It may be, the DM figured, hey, these people must be Advanced to be going here. Or, the DiveOp already screened them so hey, let's get in the water. Hence the "see you at 130 feet" briefing. I say of course, he didn't verify everyone's experience level (check cards and log books), didn't make sure everyone had done a dive plan, and specifically, didin't allow for a probably bigger novice, using more air than required.
Often, a dive guide will position themselves at the maximum planned depth, so as to make sure they are between the group and deeper water.
According to the NAUI Leadership text Page 120 "guides have a responsibility to the whole group that supersedes jeopardizing themselves (in chasing a diver who descends deeper than the planned maximum depth".
For those of you who have done a Rescue Diver or First Aid course, you know that a rescuer is not required, and is actually warned, not to compromise their personal safety to attempt to assist another diver (don't create another victim).
According to coppermaus, the DM took the whole group down to 160 feet. This would be a good point in proving some responsibility on the part of the dive guide in case things had turned out worse. However, it would be hard to prove given the DM's broken dive guage, assuming it was found to be so.
The DiveOp might be questioned as to why they allowed a relatively novice diver (the buddy) to go to the site. At least one of the divers had the necessary experience.
Since the buddy pair shared a computer, did not do a proper dive plan, and seperated themselves on their own initiative, then they would have a hard time absolving themselves from responsibility.
Please go reread your dive manual. Having a DM check your equipment after you have assembled it, check your dive plan after you have planned it, and draw a quick sketch of the dive site, are good things to do. NEVER feel embarrassed about asking for help from the DM such as "please check my NDL and air consumption calculations, or please tell me again exactly where are we going to go, how many tunnels, what if we get seperated from the group?". Rather than showing you are a novice diver, you are showing that you are a smart diver, and the DM will be quick to realize you are a safe diver. They are paid to assist you, not nursemaid you. You can bet they will step in to help you if you have any problems underwater, to the best of their ability. BUT, you should do a full buddy check (BWRAF) and make sure you stay within one or two arms length of your buddy during the whole dive.