To me, it has always been an analog vs. digital thing. Analog can equal warm if done properly, digital gets run through tape emulators to approach warm.
On the slightly more technical side, it’s where the harmonics live within the distortion of the original signal. In other words, even harmonics can add a little distortion, such as in an audiophile tube amp, without causing listening fatigue. Odd harmonics tend to add a brittle distortion. If you listen to a Krell amplifier, for instance, you probably can’t push it into distortion, but being a very fine solid state amp, it doesn’t round off transients at all, so there is a quickness to the signal that a driven tube amp, such as a McIntosh, will round off a touch. The Mcintosh, playing the same signal as the Krell, might be considered warm, whereas the Krell would be considered accurate, and therefore a bit brittle in comparison. If you want to hear bones breaking accurately, you go with the Krell. If you want warmth, you go with a tubed McIntosh. Both fantastic amps that I wish I could own, but frankly, I’d take the McIntosh, partly because of the sound, and partly because of the warmth that happens in my pants when I see glowing tubes. Sorry for the graphics, but to me a McIntosh tube amp is like love plugged into the wall.