I thought cenote meant cave. Or is it an overhang or not-quite-a-cavern or something?
Although I actually can't remember the exact translation for "cenote" suffice it to say, the "cenote" is merely a hole in the ground with water in it and it can be quite small or quite large. BUT, when people talk about "cenote diving" they are NOT talking about diving in the "hole in the ground" but, instead, the "cave" which is accessed through that "hole in the ground."
In cave diving, people have defined the "cavern zone" as that area WITHIN THE CAVE where you can still see daylight (note, there are many formal definitions, but for this discussion, this should suffice). Thus, when you dive in "the cavern" you are DIVING IN THE CAVE -- that is, you are in an overhead environment, i.e., there is no direct access to the "big air tank in the sky."
So, in summary, when you do a "Cenote Dive" you are diving in a cave but, in theory, can always see daylight from the entrance/exit (and yes, I'm writing "in theory"!). People who have only an Open Water Cert (i.e., the basic scuba training) can be led on a Cenote Tour by a guide under the current rules of good diving within the Yucatan. But those of us who go to play/dive the caves, really hope that people will be reasonable in their evaluation of their diving skills and only those who have, in fact, mastered to a significant degree, their buoyancy control and trim, will take these tours.
They are, in fact, fabulous experiences and my first one changed my diving path forever.