E.g. a violin can be bowed, allowing a single note to have more changing character than a guitar. On the other hand, a guitar is more polyphonically capable than a violin. Contrast them both with a midi trigger that only plays a single sound, for a set duration. You could say that the guitar and violin have a greater degree of freedom than the midi trigger (more possible sounds can be created), and a debatable comparison between them on how many dimensions are allowed by the instruments. Is there a word for this "degree of freedom"?
The word "mudraic" (from "mudra" = gesture) has been used by Indian philosopher P.R. Sarkar to describe how closely the gestures of the musician are connected to the sounds that are ultimately produced.
For example, the violin and similar instruments are highly mudraic, because even slight changes in the movement of arms and hands lead to significant changes in sound. Additionally, this is true for both hands, because any slight change in the left hand changes the pitch and the clarity of sound, while the right arm's movement, speed, angle, etc. all lead to different sounds.
At the opposite extreme, the organ is not a very mudraic instrument, because it is mostly a matter of turning various fixed pitches off and on at given times, with little correlation between your gesture and the sound that is produced.
The Theremin is highly mudraic for sure. The guitar, I'd say, not very much, especially when played with a pick, but a bit more so when strumming, string bending, rasgueado, etc. are employed.
Similarly, the trombone is more mudraic than the saxophone, the accordion is more mudraic than the synth, percussions played with bare hands are more mudraic than percussions played with sticks, and so on.
It should go without saying that the above doesn't imply that mudraic instruments are better or worse, easier or harder, or anything like that. This is just one dimension, out of the many different dimensions where music is created and lives.
There is also some scope for debate, I suppose: for example, how mudraic is drumming? Although any drummer could be cloned by a robot that hits the various elements at the right time in the same spot with the same force, there is no question that drummers must move a lot in very specific ways to obtain their results. My personal analysis is that drumming is highly mudraic with regard to timing, but not mudraic in the context of sound production.
One last observation: in almost every case, to play louder one must move wider, stronger, faster, but because this is true across the board, for virtually every instrument, I would not consider it a differentiating factor. In other words, even if with a given instrument you must e.g. swing your arm wider to produce a louder sound, that alone doesn't make the instrument more mudraic, simply because something analogous is the case with virtually every other instrument as well.
I might use either expressiveness or versatility, depending on the context. For instance, electric guitar volume pedals are called "expression pedals" even though they only modify volume.
I don't think you're going to find a term that's more specific to music than those, because things like timbre and polyphony tend to be talked about independently. On a synthesizer, the attributes are called parameters in the abstract, but I think it's fair to say that a synth with more parameters is more expressive.
That said, to my ear, even "expressive" probably doesn't include polyphony. So if someone says that an acoustic piano is expressive, they're probably talking about the dynamic range, rather than the harmonic possibilities. Likewise, a singer with a very wide pitch range isn't necessarily expressive.
"Expressive," then, seems to imply a precise ability to modulate pitch & volume across time. This paper calls it a "deviation from music notation," which I assume excludes expression markings. But note that such deviations, from the POV of an individual note, are called articulations. But no one says that their instruments are "articulate" :p
With that in mind, perhaps degrees of freedom is already the best term. Though I don't think it's at all wrong to use expressiveness here, especially if you signal that you mean expressions beyond what falls under articulation.
Colorfulness is also decently generic, though nonstandard.
I would say that an instrument has its own set of characteristics.
Range: lowest note to highest note
Methods of playing: e.g. plucking and bowing
Tone palette: Which harmonics are present in a given note and how these can be varied by the player
With regard to Midi I would say that you can vary the parameters; volume, pitch, duration, attack, decay, etc.