In my experience, "warmth" is caused by speakers creating a low frequency tendency in the room at the listening position. A tautology no doubt, but illustrative.
I created a "flat" FR (+/-5 dB) in my room using a GEQ. I hated it. Shrill. I then experimented with my GEQ settings using REW measurements as a guide. I found a slowly decreasing FR from about 100 Hz up more pleasing (1.2 to 1.8 dB/octave decreasing slope). The slope of the FR can be altered easily enough to create whatever sensation of warmth your ears find pleasing.I discovered much later that a -1dB/octave is recommended. Guess I like it "hot"!
So what about those systems that don't have a purposed frequency altering device. What makes them "warm" or "shrill". After all, the electronic specs are the same (let's say).
Electrical and mechanical interactions of each component with every other equipment in the system and the listening environment ultimately create the sound we hear. Want a slow roll-off of low frequencies? Choose a component (say preamp) put a big capacitor in series with the output and hook it up to an amp that has low input impedance. Boom, low frequency roll-off and attenuation if the preamp's output impedance is a large fraction of the amp's input impedance. But manufacturer's specs don't tell you this. They can't because they don't know what other components you have. You have to figure this out yourself or have an expert do it.
I enjoy tube preamps. I measure them. My AR LS22 has 0.01% distortion with and without my GEQ in the signal path. Flat FR. Same for my recently refurbished SP6B. I first paired these with a Bryston 2.5B cubed. OK. Then a Cary and then a CJ amp. Ho hum. I went back to my old grizzled ML #27. I replaced all the big blue caps. This created the sound i really liked. Why? What were the others lacking?
Current reserves!
My Revel f208 speakers are a somewhat difficult load to drive. At about 50Hz and about 2.3KHz, both the impedance and phase are low. this creates a demand for high current at those frequencies. If the amp cannot deliver that current cleanly because of limitations in the PS or elsewhere, then distortion will occur.
This is one example. One needs to be aware of how every component interacts with the others and with the environment.
Clearly, my opinion is not the last word on this topic. It is based on my experience, training (Physics), and knowledge.
In closing, "warmth" is not a distortion. It is a result of choices made. A system can have distortion that is buried in the noise floor and still sound warm.