The damage and attack types do matter since HP are so abstract. If it’s a poisoned weapon, then I would narrate it as a very small scrape. Context specific. But narrating every single hit as damaging the body takes me out of the fantasy. It starts feeling like a video game where the characters take several wounding hits without actually having to deal with the consequences of being wounded.
Generally I use the half HP threshold for physical injury, assuming physical weapon attacks. Still, HP are always super abstract and handwavey. Hit by a fireball? Why’s your cloak, bags, clothing, etc. all fine? Even when you drop to 0 hit points! Very abstract and gamey, but easy to track.