Pot-hole (in North of England where many of the long-known open
entrances are vertical shafts).
Otherwise, "cave" and "cavern" have been used in both technical
and common parlance for centuries.
Some caves have archaic individual names, if only given by
occasional travel-writers in the 16-19 Centuries. For example,
Porth-yr-Ogof in South Wales has been called things like "Porth
Gogo" but the name by which we know it now, was always the main
one. ('Ogof' in Welsh simply means 'cave')