What I mean by Warm Sounding Speaker.

I agree with the OP on this. That said. a "perfect" speaker would output simply what was fed to it, nothing more or less. I've said this before and drawn criticism for it. Audiophiles chase whatever distortion suits their ears.  

“Warm” is created by an elevated midbass and a dip in the presence range, combined with a gently sloped upper end rolloff. The bass boost acts as a sort of built in loudness compensation, as most home listeners play music below the equal loudness threshold. I prefer a flatter response but don’t object to the gentle rolloff above the midrange, as many classical recordings are too closely miked for my taste, and the tilt helps mitigate the effect. 

I like “natural” sounding sounding speakers so good recordings just sound “right” to me, which admittedly may sound warm or cool to someone else cause it’s a moving target based on personal tastes/hearing.  “Warm” or “cool” to me implies a coloration of some sort one way or the other.  That’s all I got. 

a warm speaker is:

1. …a very smooth performer, with no particular area more prominent. 
2. … should have unique tones for each performance, not color each performance with it
3. …has the possibility of sounding warm when the source is warm.  

​​​​​​​@erik_squires  You have captured the essence of a warm, musical sounding speaker.   In a recent thread, I listed attributes of the quality of musicality, which a believe applies to the quality of warmth   I have included it herewith, listed in my order of importance, and annotated applicability to warmth.

• Timbre (to me critical to both warmth and musicality) 
• PRaT that is established by micro and macro dynamics, and transient response especially the leading and end edge, without ringing. (More important to musicality, but if leading edge or pace and rhythm is excessive, it can reduce natural warmth.  If trailing edge is excessive it can muddy the sound add to excessive warmth). 
• Detail where most important is the ability to reproduce harmonic decay and hall (or studio engineered) ambient effects.  To me, this contributes to the quality of bloom and natural dimensionality.  I find this the area where there is the greatest difference in listening to a live acoustic performance in a good hall where there is a detailed, sweat tonal characteristic emanating from a point source and expanding and decaying into the venue as the concentric waves generated from dropping a pebble in a still pond. 
• A natural, sweet, and liquid midrange.  (The ability to accurately reproduce harmonic decay and ambience is also critical to natural warmth)
• Highs without harshness or sibilants. 

​​​​​​​In my discussion on musicality image density, which also applies to the quality of warmth.

 @fastfreight +1 Vivid Giya.  I have the G3.  

Interesting that you bring up warm with respect to a speaker. The term to me does not have the same implication as it would electronics. To me warm speakers are unnaturally bassy... probably rolled off... not a good thing, ever. 

But in the electronic realm. The advent of solid state most early electronics were lean, trebly... midrange poor. Then the CD was introduced. Then they figured out how to supply high current in amps and preamps... but still left the midrange lean. This was at least a 40-50 year problem. Now it is mainly a question of balance between details/slam and a balanced sound and warm is good as in midrange bloom.