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 say outdoors. Indoors has a vibe of being human made to me, and a low likelihood of encountering wildlife. So I guess walking around a mine is indoor cave exploration
Many caves are modified to accommodate their visitors. This could be a small as leaving behind oxygen tanks or equipment, or as advanced as installing electrical wiring and ventilation. This was the case of several caves I visited as a tourist with no real exploration skills.