Terminological Exactitude....

It's true that we'll never agree on just what 'warm' or 'analytical' mean, in a descriptive sense, due to how we hear and interpret it ourselves. There's no consensus due to our individual take, our connotations, on what we ascribe to the term.

It's not like how we can all agree on how the sun feels warm on our face or how water feels as it flows between our fingers. Just the mention of it brings familiar sensations that are rather plain and easy to imagine.

Due to the nature of this hobby (affliction?) we tend to hold closer to terms of our own choosing which are backed by a strong sort of attachment to the term as we try to define it. Any misinterpretation or conflicting realization of the term leads to conflict and debate. There are times when it seems apparent that disputes are agreements, save for the description. Funny that.

To me it helps when describing something in a poetic way as it leaves so much room for interpretation and yet, agreement. The flip side is that some get rather worked up over it. Until we can all agree on just what means what, we'll just have to give everyone a little more latitude.

All the best,
Nonoise

If all sound, including music, consists of two physical properties, namely amplitude and frequency,

All sound comprises of phase & amplitude. Further, time & phase are related in that time is phase. It takes a certain time for the sound originating from a source (any source) to reach your ear. Depending upon how much time it takes dictates what the phase of the sound is (w.r.t. its originating point) when it reaches your ear. Phase changes from zero to 360 degrees & starts all over again from zero infinitely until the sound dies out. Amplitude of sound falls off 6dB per doubling of the distance. So, the further away you are the lower the amplitude.
In nature, phase is the independent variable & frequency is a derivative of phase. So, if phase changes, frequency will change. And, when the frequency changes, the timbral aspect of the sound changes.

then one could argue that much of the audio language we use is vague, and sometimes extremely difficult to understand.

Continuing what was written above, the audio language is basically describing what the distortion sounds like upon playback.
As you know, all electronics & all speakers create distortion. The name of the game is to identify components (electronics + speakers) for any given price-point that playback music with the least distortion. That will always enhance one's music playback session(s).
So when people write

For example, what are we supposed to understand by words like 'analytical' or 'warm'?

they are really describing the sound of the distortion - specifically the phase distortion - thru their resp system or some system they auditioned (friend's place, dealer's place, etc).
it's important to remember that when the phase of the original signal (coming off a LP or a CD or a music stream) is altered (by electronics or the speaker), the frequency content of that music playback session is altered. This in turn changes the timbral accuracy & this completely alters the music for the worse (in general. there are many people that like phase distortions because it gives them a coloured sound they are looking for specifically).

Once you understand this, it is very easy to understand the audio language.

Bomboywalla; yes indeed, phase distortions appear to be one of the keys to understanding this.
I recently had the pleasure of meeting one of the more influential power supply designers, whose researches are specifically dedicated to analyzing this phenomenon. His studies lead to interesting conclusions regarding phase shifts far above the conventionally held limits of audibility, and their effect on what we seem to hear. Fascinating.

And I do agree with Nonoise; giving latitude is essential. No need for conflict!

Nonoise & 57s4me,
re. giving each other some latitude: yes, I'm in favour of this because what we want on this forum is discussion & not a fight even if the discussion is outside our resp. knowledge realm & outside our comfort zone. The whole idea of forum participation is to expand our resp. knowledge base altho' I know that not everyone here subscribes to this line of reasoning.
And, no, I'm at the same not in favour of giving others latitude because if the definitions are not clear we have different people interpreting the same thing differently & the advice given is all over the map to the point where it is useless.
So, if I don't understand what your definition of "warm" or "analytical" is then I could off in left field by applying my definition & drawing wrong conclusions & giving bad advice. And, I've found this to be frustrating on several occasions.
So, if everyone was diligent in starting off their verbal descriptions with their definitions then we could cut each other a lot more slack & even be on the same page. But this seldom happens - I've to depend on my knowledge of "warm"/"analytical" & hope that the OP shares the same opinion.

Wow i did not know that.