You could add to that list many other European countries as well, e.g. Spain, Italy, Hungary and Norway.
However, there are also countries without any natural caves at all, e.g. Finland. Instead, we Finns go cave diving in mines. There are about half a dozen mines that are regularly dived. A mine called Ojamo is the best known one. It has diveable mine tunnels ranging from 20m/60ft level all the way down to 238m/700ft depth.
The minimum dive certification for mine diving is "Intro to Cave" or "Mine Diver", but you may consider continuing to "Full Cave Diver" or "Technical Cave Diver". Also rebreather training is one consideration.
Personally I do not know Bonne Terre, but based on their web pages, it would seem to be open water guided theme park diving in a mine environment with direct access to the surface at all times. Minimum 12 years and open water C-card. Likewise based on respective web pages, LaMotte seems to have real mine diving for suitably qualified divers i.e. proper cave diving certs.
Mulla, you might want to contact the Cave Divers Association of Australia for further information about training and dive sites near you.
Cave Divers Association of Australia | Incorporated in South Australia