Thanks for the detailed feedback! I made the fixes and adjusted the wording for now while I explore hosting a server.
Checked the new build (19964356) and both are fixed. With shared light off, the intro now gives the solo version ("Solo walk: your torch and the keeper's lamps.") and the strangers line only shows in shared mode. I loaded it at the same moment in Chicago, London, Tokyo, Auckland and Honolulu, and all five got Cave No. 4 (Fri, Oct 23). Nice touch telling each player when the next cave drops in their own time (7:00 PM Chicago, 1:00 AM London, 1:00 PM Auckland, all the same UTC midnight). 0 console errors. Good luck with the server.