First, thank you for moving this to meta! The issue is that the math educators stack exchange is a forum for discussing mathematics education, pedagogy, tactics, ideas and research (that is education research, not math research). Here, you're pointing out interesting linguistic facts about mathematical naming conventions, but it's not at all clear what they have to do with education. Unless you spell out clearly how your linguistic theorizing related to teaching students, this isn't a good question for the math educators stack exchange.
On a side note: I think you are also getting some push back because you are giving the impression that mathematicians don't think about language in a logical way. As an example, you use
"Formal definition: A representation U(G) on V is irreducible if there is no non-trivial invariant subspace V with respect to U(G). (Wu-Ki Tung, Definition 3.5)
My translation: When a representation on a space is reduced to the point that only that space and {0} are its only two subspaces that can hold their vectors from being pulled out, then we have an irreducible representation."
The formal definition is a very precise logical statement. Your translation is not a completely incorrect (and incoherent) description of the mathematics, it doesn't even respect the logical (propositional) syntax of the sentence. It makes it very hard to understand then what it is you're proposing.
If you're proposal is that we should first describe mathematics propositions in a way that a laymen would understand them your in luck! There is a lot of pedagogy on that very idea. If you're interested in having a discussion about how we teach students logical prose I would suggest doing a quick google search, reading a bit about the state of the field, and posting about your questions related to education. If you're interested in discussing how linguistic structure can be translated into logic, I would suggest posting to the linguistics stack exchange.
If you're interested in why mathematics is written the way it is, I would suggest reading (and writing, having critiqued!) a lot of higher math. What you'll find is that it's actually hard to be more clear than the formal definition above.