Electrical wiring issue with Ring Camera
Connect red to black here
I'm betting that whatever they did had the photocell on this light switching the other light on and off.  (You may be able to verify this through some logging facility on the old Ring, as it'd have been turning on at night and off during the day.)  So, I'd tie the red wire here in with the black wire and the new Ring's hot wire, and see if both Rings power up at that point.