In some cases, the cognates of onomatopoetic sounds are highly similar even across unrelated languages. In these cases, the sounds of words seem to be an attempt to echo naturally occurring sounds. Other words follow etymology stemming patterns. Still others have no known relationship to what they attempt to express, or to existing expressions; I'm in fact not aware of any, but pretty sure they exist.
No, there is no terminology for the level of closeness a 'phonetic expression' takes to its 'meaning', quite simply because this is not something that is researched or discussed in linguistic circles, as there is no real reason to engage in this sort of enquiry. Here are some reasons why:
1) As addressed in the comments, and as widely believed throughout linguistics, most words are arbitrary with regards to their meaning, with no real connection between the types and combinations of sounds in the word to the referent.
2) When onomatopoeia does occur (and it is rare, in that most languages have a tiny number of onomatopoeic words compared to arbitrary words), it is essentially formalised mimicry. As has been pointed out, you can't get very far comparing human vocalisations to non-human sounds, whether they are sounds of animal calls, falling rocks, rushing water, or whatever. All of these real-world sounds will be produced by very different means, and humans can, at best, come up with an approximation of them that works with the constraints of the human vocal tract. More importantly, an onomatopoeic word can't just try to imitate a referent sound as precisely as possible - it also has to abide by the phoneme inventory, phonotactics and phonological processes specific to the language, at least to a certain extent. This is the main reason why onomatopoeic sounds vary cross-linguistically, plus the fact that different speakers may have different perceptions of the same referent sound, and may therefore be mimicking different aspects of it.
Furthermore, an onomatopoeic word is, by virtue of being formalised, a word like any other, and subject to the same processes of language change, in particular sound change. This will also account for a lot of variation in onomatopoeic words for the same referent, and something which may have originally been a close approximation of a referent sound is likely to become further and further removed from it.
3) There is an active area of research into sound symbolism, but the findings are in general tenuous, and are certainly not based on direct correspondences between a word and a particular sound or thing it references. Rather, research on sound symbolism investigates the more subtle connections between certain types of sounds and certain types of very broad, non-concrete meanings. E.g. sonorants as opposed to obstruents, or voiced sounds as opposed to voiceless sounds, or back vowels as opposed to high front vowels, and the connotations they seem to have of qualities such as 'spikiness' or 'roundness', or 'sweetness' or 'sourness', or 'tiny-ness' or 'big-ness', and so on. There is an accessible overview of such research in this New Scientist article. Phonestheme research falls under this category, too.
Sound symbolism research has found some very interesting patterns based on human interpretations of nonsense words (and this has produced a wealth of knowledge for advertisers coming up with new brand names and slogans), but this is not so straightforward as an easily observable connection between the sound of the word and the referent, and of course there is still no terminology to describe the degree of similarity between the two, because the referent doesn't generally have a sound in many examples of sound symbolism. (What is the sound of 'sour'?)
The terminology for how much a word sounds like its meaning is pretty simple:
- onomatopoeia - a word or short sequence of phonemes that is imitative of a non-linguistic sound. for example the sound of a bee is called 'buzzing' in English, the voiced alveolar fricative evocative of the drone of the bee. In comparison to all the sounds humans make, these sequences are very rare.
That's it. In fact the vast majority of linguistically produced sequences are so arbitrary and so distant from imitation, that there is no specialword for the larger set of non-onomatopoeic words. And there is no real 'in-between' stage of 'somewhat sounding like the thing it stands for, but not quite', because onomatopoeia includes that too. For example, in a bee buzzing, have you ever heard a voiced bilabial stop? That is 'in between' but is still considered onomatopoeia.
No, there is no specific term for how well an onomatopoeic word approximates the sound it refers to, because there is no way to measure how alike two sounds are. Remember that nearly every utterance made by the human vocal apparatus is a combination of several articulators sounding in unison (or phased relative to each other), making their output even more difficult to compare to any given sound in nature.
I may have misunderstood what you're looking for. But if I understand your question right, there is a term for what you're talking about: iconicity. In cognitive linguistics, expressions that are similar in form and meaning are said to be "more iconic" than expressions that don't show that sort of similarity.
So for instance, an onomatopoetic animal name like cuckoo is said to be "more iconic" than a non-onomatopoetic one like eagle. If you say something is lo-o-o-o-o-ong, with a lengthened vowel, that's "more iconic" than saying very long. If you say he did it over and over and over, repeating the word to indicate a repeated event, that's "more iconic" than saying he did it many times. Saying he went home and fell asleep, with the events described in the same order in which they occurred, is "more iconic" than saying he fell asleep after going home.
Still, I agree with all the caveats that other folks have given you. It is impossible to quantify exactly how iconic an expression is — there's no, like, "iconicity quotient" you can calculate. And often it's not even clear which of two expressions should count as more iconic than the other. Is cuckoo more or less iconic than whipporwill? Or for that matter, is eagle more or less iconic than pigeon? There's really no way to answer.
So as a concept it's not always very useful, and it's certainly not very precise. Still, the concept is there if you want to use it.