I thought the charts were the same for any kind of light. But I'm noticing in halogen the calling of 4000 is white light and a 6500 a very noticeable blue. But in LED a 6000 to 6500 is white. I'm really confused here.
Having come from photography to the world of flashlights, I am a bit confused myself. For matching midday sunlight, 5000K to 5600K is typically considered to be neutral in photography.
I suspect that part of this is just marketing hype. Nobody want to advertise their "cool white" flashlights as "blueish." Part of it is also historical artifact. Early LED flashlights were almost exclusively cool-white, with the accent on cool. From what I have read, many had horrible tints. Nevertheless, they were called "white" or "cool white." Now we are stuck with the label.
I don't know a thing about headlights for automobiles.