I was just in Coz in January and was able to do a cenotes excursion one day.
I asked the same question and they do refer to it as 'cavern' vs. 'cave' diving. I can tell you, however, that during the two dives I made there were stretches where it was dark and you needed a light, plus the surface or exit was not always visible. If you were really, really, really claustrophobic, you may not be comfortable during some portions of the dive. Not that you were in tight places where you could get stuck, but it did get dark and felt like a night dive in spots. If you freak out at all diving in the dark, you may want to pass. You may be in the dark for 5 minutes at a time depending on which cenotes you dive.
With that said, I felt completely safe and comfortable the entire time, and I am one of those tight space claustrophobes. The op that ran my dives (Dressel) was great.
Maximum of 3 divers per instructor/guide.
You stayed behind the guide and over the permanent guide line in single file about 6 feet apart.
Every diver is required to have a light (take your own along, the loaners sucked)
The guide wore doubles and had an extra long octo in case anyone had air issues during the dive.
The dive was handled like a wreck or cave dive with the rule of thirds .. 1/3 of your air in, 1/3 out, 1/3 for emergency. First diver in group to hit 1/3 air, the entire group turned around.
I noticed that unlike diving at Coz where the AL80's rarely seemed to be pumped up more than 2750psi, the two tanks for the cenotes dives were both over 3000psi for all divers. Nobody in our group had to call a dive early because they hit 1/3 air used before reaching the normal turn-around point. Not sure if they took a little more care to get a full air fill for the cenotes divers, but I can be a hoover and appreciated the 'extra' air.
I had a great time and would recommend that if you have the chance, at least do it once in your lifetime.