Here's how I like it.
Hit points represent your ability (stamina, luck, morale, fortune) to avoid the terrible thing that's about to happen to you.
When you're out of hp, the terrible thing happens.
What that means depends.
This means that when when a trap "does 8 damage to you," it doesn't actually do any real damage at all unless it took you to or found you at 0 hp.
This means a "hit" is usually just a close call, something that takes serious effort to dodge, block, or avoid, or something that breaks your spirit or gets under your skin.
This means you need to know exactly what the hit was attempting to do so that you can narrate the effect properly.
If you wish, you can employ saving throws after hitting 0 hp as a sort of luck-mitigation: yes, you're chopped in the legs, but it only breaks your leg rather than severing it. Either way, you're badly injured.
So saving throws become a way to determine the degree of badness happening to you, where the degree should seemingly be random.