I've heard this argument before, but I don't agree to the degree you're using it.
There is definitely a distinction between "formal" language and informal, and people will judge the quality of writing by the degree that it conforms to accepted practice in that field. When I went to law school, our first semester writing class was focused in large part in learning the arcane and quite frankly stupid citations rules that our profession uses.
But the problem is, these aren't really what our profession uses. They're kind of the basis, but really they're the rules used by law schools and journals. Courts use similar rules, but have their own local flavor. The Supreme Court, for example, has different rules for italicizing certain things than the Blue Book (the standard legal citation manual). But so what? It's not like the issue of whether the word "see" is in italics suddenly changes its meaning or makes it harder to understand. This also ignores other things like the fact that the editors of each successive edition of the Blue Book like to change things just to make their mark, rather than because there's a reason to.
Language is only valuable to the extent that it's understood, and that relies in large part on standardization. We all have to be on the same page. However, this often gets taken too far. I'm having an incredibly difficult time imaging a scenario where either of the two examples you used would create any ambiguity. I get that a term of art for a given field would look weird if spelled differently, but that just goes back to the standardization thing.
Too often, I think, this kind of thing becomes a way to separate people, and can overtake the underlying ideas. So we need to be wary of creating arbitrary rules for rules' sake. The "don't end a sentence with a preposition" rule is a perfect example: this was made up by teachers in the 19th century to mimic Latin grammar, simply because they wanted English to be more in keeping with Latin, and therefore better somehow. It serves no purpose, and can often actively inhibit clarity, requiring the writer to go through all of those "with which" or "of which" contortions to match.
I'll admit this is kind of a bugaboo with me, because I think far too often it's a form of intellectual snobbery. I also think it cheapens communication; some of the richest literature deliberately ignores convention. I mean, can you read Cormac McCarthy and say his books would be better if he used basic punctuation? I also think it's quite arrogant. I wouldn't try to mimic the English of the inner cities in a courtroom, but that doesn't mean that version of English is "incorrect" overall.
So again, your writing/speaking distinction is not enough in and of itself to justify such arbitrary rules. The question is whether the thought is clear, full stop (which can include conforming to standard terminology, to be sure).
I want them to know what correct, concise, unambiguous writing looks like so that they can at least have choices.
I'm sorry, but this mindset drives me absolutely crazy. You're not the Emperor of English! No one person gets to impose their ideas of what a language "should" be by fiat. You can say what the conventions are, but I think it goes too far to say that this is "correct," as if there is some objectively True English out there. Language by its nature is not objective, and saying that it's "correct" just comes across, to me, as condescending. Just as you're trying to get your students to communicate clearly, it's important, frankly, what words you use.
My Spanish-speaking student told me that in her country, they like to write long sentences with little punctuation. She said that they were taught to almost never use commas. She also said the writing is tedious and hard to understand.
That's strange. I mean, I agree that Spanish writing definitely favors run-on sentences to a greater degree than English, but plenty of languages do this...Ancient Greek writers would go a full paragraph with only one active verb. It's different, sure, but it's ridiculous to say that it's per se hard to understand.