I've never heard anyone claim it is about openings to the surface. The difference between a cave and a cavern is a matter of size, cavern connoting a large cave. Webster defines cavern as "an underground chamber often of large or indefinite extent."
A cave, on the other hand, simply is a "natural underground chamber or series of chambers."
Note that cavers (cave surveyors, mainly) have imposed different definitions on what a cave is for purposes of record keeping. In some states - Ohio - a cave is any natural opening large enough for a human to enter. In other areas, it requires at least 50 feet of passage to be listed. And yet others require a zone of darkness.
No matter the size of the cave or cavern, they still are pretty cool places.