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 agree with the door argument but would also add my two cents… If it’s a cave dwelling it would be inside but you also wouldn’t be cave exploring. You’d be in a ‘civilized’ area