Terminological Exactitude....

Are you an engineer or something similar?

If it sounds good and it measures good, it is good. If it sounds good and measures bad, or if it sounds bad but measures good, we haven't learned exactly what to measure yet.

Progress is never ending. It would be foolish to think that mankind now knows everything that there is to know. We are constantly evolving, and learning new things everyday.

Maybe one day measurements will be more accurately able to predict what we hear. Maybe one day measurements will be more accurately able to predict the weather. ;^)

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.