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Cooking Up Contingency Plans
(i.imgur.com)
Since my polymorph meme has only garnered three downvotes so far I thought I'd offer a bit more controversial take, and see if I can manage to stir the pot a bit with this one.
You trigger after you take damage but before you fall unconscious, like numerous other triggers in the game. I just used the Strength Before Death example because it shows even without magic there's enough time to squeeze a whole ass turn in between those two events.
I fail to see how that's supported by RAW. If damage reduces you to 0 hit points and fails to kill you, you fall unconscious.
The fact that a Fighter subclass gives you an ability to act otherwise doesn't affect this, any more than you could say you stay conscious at 0 because Zealot Barbarians can do it. You don't have that ability. You're unconscious or dead when Vampiric Touch triggers. Vampiric Touch fails.