Snake and serpent mean exactly the same thing. But they're different words when they're treated as derivations. The obsolete brass instrument is a serpent but cannot be called a snake. The plumber's tool is a snake but cannot be called a serpent. (Source: Mark Rosenfelder)
Are there any kind of known rules for these kinds of derivations?
I'm aware of a huge project conducted by Borer (2005a,b; 2013), a major part of it is dealing with word polysemy. However, her project is not purely morphological per se, since you requested theories that deal with polysemy, I believe, from a syntactic point of view, that even some types of polysemy (especially systematic) are dealt with in the syntax (narrow syntax precisely). I recommend reading her last book published in 2013 (Structuring Sense Volume III).
The theory being developed in Borer's project is purely constructionalist (contra lexicalist positions).
There is no derivation in the example you gave. A derivation, in Linguistics, is when a morpheme is added to another morpheme to produce a new meaning.
The case you are talking about can be analysed as a case of polysemy. The morphemes Serpent and Snake are polysemic and share synonym semes at the same time.
This polysemy is due in your case to a semantic change/shift. Probably, the spring of this shift comes from a cognitive operation such as the metaphor or the metonymy.
Snake and serpent mean exactly the same thing.
Is that really true? Of course they both denote the same class of animals, but that doesn't mean they mean exactly the same thing. They have slightly different connotations, as well as different word usage patterns.
Let's start with the latter, because it's simpler. First, there's a very slight difference in register. Centuries ago, only educated people usually said "serpent". Nowadays, most speakers have both words, but "serpent" still sounds a tiny bit more formal and classy. Also, even after music was no longer described in Latin, it was described in Italian, so "serpent", being a Romance borrowing, sounds a bit more "music-ish". And finally, there are some contexts where one just sounds better than the other—the most obvious being metrical poetry, where the difference between one and two syllables matters. Third, they differ in derived forms—"serpentine" is a common word; "snakelike" feels like an ad-hoc compound that you should maybe spell with a hyphen.
But also, pairs of synonyms almost always have different connotations. People don't always consciously know what the connotations are, and a dictionary (especially a small one) may not even mention them, but they're there, and usually consistent across most speakers of the language, and they can be teased out. For example, if someone, for no apparent reason, suddenly betrays you, you might yell "You snake!", but you probably wouldn't yell "You serpent!" Compare enough distinctions like this, and the picture starts to arise: Both snakes and serpents are treacherous, but for a serpent, being devious and subtle is central, while for a snake, subtlety isn't even required, and dangerousness is central. Where do those connotations come from? It might be centuries of different word-use choices, or register differences, or sound symbolism, or the influence of common idioms and cliches… But regardless, they are there.
Both of these factors affect metaphorical extension. When you're only using the shape in the extension, it's probably more about the register than about connotations: a plumber's tool uses the lower-register word and an orchestral instrument uses the higher-register, and more-Italian-sounding one. (It may also be influenced by the military serpentine, a kind of cannon.)
On top of that, even if two synonyms really were 100% interchangeable, which metaphorical extensions take hold in widespread use is largely arbitrary. Sure, maybe either one of them could have been extended to mean the plumber's tool, but so could dozens of other words, and "snake" is the one that happened to stick. And, once it has, that makes it even less likely for "serpent" to be extended the same way, because we already have a name for the tool, so there's no need to coin a new one.
Snake and serpent mean exactly the same thing. But they're different words when they're treated as derivations.
That derivation is a misnomer has already been pointed out, so let's ignore that.
The second sentence still sounds weird to me. It seems to imply hat serpent and snake would be the same word if their meaning was identical in every context. But they can't be, as their phonological form is different, and words are complex units, made up of phonological, grammatical and semantical properties. (Regarding grammatical properties: In languages other than English, nouns might belong to different declensions or have different genders.)
So the words are always distinct because of their phonological (and possibly their grammatical) properties.
Assuming their meanings were identical at some point in time: If a plumber invents a new tool and says 'I dub thee snake', synonymy would be gone (as soon as the term passes into general usage).
Given the facts you outlined in your question, you either have to assume that the meanings of snake and serpent are not identical because one can be used to refer to a plumber's tool and the other can not; or that there are actually two words snake1 and snake2, with the second one referring to the plumber's tool and serpent merely being synonymous with the first one (referring to certain reptiles).
Yes, we know about this aspect of language. The basic idea is that of the dictionary. Part of our understanding of language is our understanding of words. Words have derivative properties from their composition from morphemes and phonemes, but also they can have idiosyncratic properties, as your clever example shows. We know about words.
