Well....I'd regard Cavern class as the first "real" cave class as that is where nearly all the basic skills are taught.
Intro to Cave and Basic Cave are the next step respectively for the NACD and NSS-CDS certification organizations.
Intro expands those Cavern skills into actual practice diving beyond the cavern zone. The certification limits are designed to limit depth and penetration, with the understanding that an Intro / Basic diver will have less distance to exit and more time and gas available to address any problem that might develop on the dive. Similarly, the restriction to no jumps, gaps, tees or other navigational decisions also reduces the issues an Intro diver would have to address on exit.
"Cave" and "Full Cave" are the next steps in the NACD and NSS-CDS progression and this certification involves making jumps, gaps and navigating tees to complete circuits and traverses, and it involves, much longer dives to thirds or close to thirds for gas planning, and in N FL usually involves decompression.
This can be broken into two steps, with the diver completing an Apprentice level, which is a level, not a permanent certification, that allows limited navigation while the diver builds more experience, before coming back to complete the Cave or Full Cave certification.
Most of the courses beyond that (Cave DPV, stage diving, trimix, etc) usually don't occur until the diver has completed at least 100 cave dives at the Cave/Full Cave level.