You’re indoors in the sense that you’re protected from the weather and the elements, and the cave could even have some kind of covering or entrance area that could be considered a door or doorway. People have built homes in caves.
Is caving an outside, inside activity?
I’m going to say outdoor.
The “door” part doesn’t really have any significance. No one would say camping under the open sky is an indoor activity, even if there’s a fence with a door around the campsite.
I think it makes more sense for the deciding factor be whether you’re in a controlled or uncontrolled environment. And while part of the cave might be controlled if there’s an artificial entryway or home, that’s not what you’re there to see.
Maybe I’m gatekeeping, but isn’t a fence door a gate?
I’ve been in show caves in Bonaire that had some degree of climate control, mainly for humidity and to ensure sufficient oxygen for visitors for a comfortable experience and to accommodate large groups.
Certainly that’s the exception rather than the rule in my case.