Super simple death save rule: (TL;DR: check for changes to the RaW in bold italic. )
When you drop to 0 hit points, you either die outright or fall unconscious, as explained in the following sections.
Instant Death
Massive damage can kill you instantly. When damage reduces you to 0 hit points and there is damage remaining, you die if the remaining damage equals or exceeds your hit point maximum.
Falling Unconscious
If damage reduces you to 0 hit points and fails to kill you, you fall unconscious (see appendix A). This unconsciousness ends if you regain any hit points.
Death Saving Throws
Whenever you start your turn with 0 hit points, you must make a special saving throw, called a death saving throw, to determine whether you die or hang onto life. Unlike other saving throws, this one isn’t tied to any ability score. You are in the hands of fate now, aided only by spells and features that improve your chances of succeeding on a saving throw. Roll a d20: If the roll is 10 or higher, you succeed and become stable. Otherwise, you fail and die. Rolling 1 or 20: When you make a death saving throw and roll a 1 on the d20, you are really really deader than dead. If you roll a 20 on the d20, you regain 1 hit point. Damage at 0 Hit Points: If you take any damage while you have 0 hit points, you suffer instant death.
It’s fast and simple, and allows for the “left for dead” trope. If you add or gain an exhaustion level to the damage at 0 hit point clause, you can also die of extreme heat/cold .
Variant: Ye Olde System Shock
Instead of a death save, you roll a good old system shock, OD&D style, with percentile dice for nostalgia sake. Your system shock value is 40% + your CON score x3 (max 99%). You must roll under your system shock value to become stable, otherwise you’re dead.
This makes the “death save” more survivable. Perhaps too much, so…
Variant: Constitution save
Instead of a death save, you roll a CON save, DC = 10 or half the damage you take, whichever is higher.
This makes CON save proficient classes more survivable and death by critical hit harder to survive
[edit] ...or what OB1 said two posts above...