Since I'm enjoying the different rules shared here, here's a (from memory) rendition of the Fate RPG rules on encountering lethal amounts of damage.

DM and player discuss and assign an appropriate and interesting condition that moves the game along. That condition may be "dying" or could be something more interesting.

Players and the GM can invoke the new player condition to gain benefits and make other rolls easier or harder. (The core FATE rule.)

Weirdly, this covers a lot of interesting cases really well:

• the GM can invoke "dying" to keep the dying character from monopolizing the remaining combat in un-fun ways, and make it (taking lethal amounts of damage) have an in-game cost.
• the GM can invoke the "dying" condition in other ways to nudge players to find a way to lend aid ( like granting a character "encumbered" while they carry the "dying" character around)
• the "dying" player may be able to invoke "dying" as an "I'm very motivated" bonus if they're doing something very in-character that matters to their character
• "unconscious", "prone", "mostly paralyzed", can be a useful on-and-off conditions to represent recovery rolls that go badly

For GMs running a game of FATE, I recommend watching the "The Princess Bride", which milks the "dying" condition for interesting moments, in many delightful ways.