In the Yucatan most divers consider there to be a difference between 'cavern diving' and 'cave diving'. Cave diving requires extensive training and is very dangerous. In cavern diving you stay within the naturally illuminated part of the cave and, while there might be something overhead of you at periods, you're not functionally farther from a safe exit than you would be at 130ft depth with no overhead. Of course any overhead obstruction is an added risk but that situation happens in regular recreational diving too.

It wouldn't surprise me to hear that some tour operators bend the guidelines but the diving I did there felt appropriately careful.