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 would consider a cave to be “outdoors” unless maybe somethings been done to it to make it into a dwelling. Since there is no door it’s at least somewhat open to the elements, you would have to contend with various wildlife that may live there, the floor would be either dirt or bare rock, these are all things one would have to deal with while doing “outdoor activities”