I’ve seen 4000K labeled “cool white” on a screw-in bulb, “warm white” from Armytek and a few small makers, and most often “neutral white”. These are marketing terms that don’t have fixed definitions.
A poll like this was conducted last year with actual color temperatures and 5000K won with strong showings from 4500K and 4000K. The plurality voting method used there is suboptimal with more than two options though. The poll for the FW3A color temperature using a ranked Condorcet method produced a similar result with 5000K on top, 4000K-5000K being popular and options far form that range being unpopular.
Polls have also been conducted on reddit and CPF using approval voting with the popularity peak covering the same range. (4000K won overall on reddit; 4500K on CPF)