If an attack causes additional damage after a saving throw, is it counted as a separate source of damage for the purpose of failed death saves?

If an attack causes additional damage after a saving throw, is it counted as a separate source of damage for the purpose of failed death saving throws? In other words, does it cause another failed death save, in addition to the failed death save(s) resulting from the initial damage?

For instance, consider the wyvern. Wyverns have a Stinger attack:

Stinger. Melee Weapon Attack: +7 to hit, reach 10 ft., one creature. Hit: 11 (2d6 + 4) piercing damage. The target must make a DC 15 Constitution saving throw, taking 24 (7d6) poison damage on a failed save, or half as much damage on a successful one.

If a character at 0 HP is hit by the Stinger attack, they will immediately take some piercing damage. Then, they would ostensibly have to make a Constitution saving throw, taking some amount of poison damage either way.

If the wyvern is within 5 feet of the unconscious (and presumably prone) character, then its attack will be made at advantage. If the wyvern’s attack hits the character from within 5 feet, it will automatically be a critical hit, which will cause 2 failed death saving throws just from the initial damage.

Then, if the poison damage from after the Con save is considered a separate instance of damage, it will cause a third failed death save, allowing the wyvern to kill a character that is at 0 HP with a single attack. (The wyvern can also use Multiattack to attack with its Bite and then its Stinger, meaning even a character that hasn’t gone down yet could be rapidly killed in this way.)

Am I correctly interpreting the rules?

I was unable to find anything in the Sage Advice Compendium commenting on how damage done after a saving throw (as a result of an attack) is treated for the purpose of death saving throws.