I think you hit the nail on the head here: assessing how vital the equipment is to continuing the dive. In the case of your cave divers, lights are absolutely essential. So a failure means the dive should be aborted.Perhaps there's different considerations if you're a DM somewhere and your backup is there specifically to finish a dive for paying customers. My cave diving buddies with 3 or 4 lights abort no questions asked if 1 fails.
It seems that there's some disagreement as to whether or not a dive computer is absolutely essential for all dives, and perhaps this is where the dividing line lies: my scenario above, a shallow dive for which the DC would only be used for time and an NDL that exceeds my air... maybe the DC is not essential. But certainly a deeper dive the DC would be more relevant.