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?
Pedantically speaking, I have never been in a door. I’ve opened doors and passed beyond the portal they were blocking, but I have never phased into or been sealed inside of a door. :p