Transparent, Fat, Warm, and other descriptive terms:
These adjectives have such a wide range of interpretations and usage that they can be quite misleading and confusing. Even knowing that, I'm still guilty of using those words with varying standards and meanings, as much as anybody else. It's very difficult to describe tone or other subjective perceptions with any accuracy, in large part because everyone reads words like "thick" or "clean" with their own personal interpretations and assumptions. This article will not solve that problem, but I will try to give some explanation of how I understand and use these terms. Regardless of my opinions, you will always have to read these things in context, and try to understand the writer's intention. Sometimes their only intention is to sell you something, so bear that warning in mind. Advertising text is very often just a bunch of fluff and buzzwords that excite buyers, without having any literal relationship to the product being sold.
"Transparent" is ideally supposed to mean that there is zero change in your signal from the input to the output. A straight wire! Of course the reality is that every processor changes the signal slightly, and a lot of the time we want some "improvement" in the signal, so no processor is 100% perfectly transparent. This means that manufacturers and reviewers use the word transparent to mean "it does not change your signal in an undesirable way" or "it sounds like your original signal only better". Generally speaking, you can usually interpret claims of transparency to mean that the device is meant to be as uncolored as it can, aside from whatever processing it is supposed to do.
I use the words "clean" and "clear" to mean essentially the same thing as transparent, though I may especially choose those words if the signal has very low noise or very good articulation (no "smearing" of the sound).
"Warm"... That's a tough one. There is no one correct or ideal meaning for this word. Because it is most often applied to vintage tube and transformer-based gear, warmth can be thought of as resulting from mild distortion (soft clipping or increased harmonic content). Additionally, people often describe deep vocal tones as warm, and in general a tone that emphasizes the low mids will be perceived as warm. So I'll go out on a limb and say that warmth means those two things, mild harmonic distortion and/or low-mids emphasis. Feel free to disagree though.
"Fat" and "thick" are a lot like "warm", and I think they come from the same causes. To me they imply a stronger, more exaggerated effect or character than warmth, but once again there is no one right answer. Fatness and thickness may sometimes include muddiness, a smearing of the articulation of the sound.
Just for reference, I should also mention something about "distortion". Most musicians will think of a hard buzzy clipping sound when they see this word used, but from an engineering perspective "distortion" means any time the audio wave shape at the output is different from the wave shape at the input, by even a small amount. So in that sense any effect you might use is technically distortion. The reason this is important is that distortion is the opposite of transparency, so when you listen to a processed signal and it doesn't sound completely transparent to you, the question to ask is "what kind of distortion are you hearing?" That way you can start to figure out which pieces of gear alter the harmonic content, and which ones compress or clip the signal, and which ones have an inherent EQ shape. Because fatness and warmth are both caused by distortion, you can see that distortion is not always a bad thing--in fact sometimes it is highly desirable.
I'll add to this list as I think of more of the vague and arguable words that are used to describe tone.