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.