Between or among

I learned similarly (although around a decade later). But between has become a go-to for any number, probably because to a lot of people it just sounds better (maybe the long e in the second syllable?).

Funny, it just grates my nerves when I hear what to me is incorrect language usage. However, I am not a grammar expert. Thanks for your input. I feel vindicated. I have other grammar things that bug me. As I encounter them I'll ask about them.

I found this article on your question quite interesting because I had, as well, been taught as you. However...

"If you're at all interested in grammar, then you've probably heard the maxim about among and between: only use between when you're referring to two things ("just between us," "between a rock and a hard place," "you must choose between cake or death"), and use among when referring to more than two ("you're among friends," "among the many options available to you"). But like many maxims about grammar and usage, this isn't necessarily the case.

History of 'Among' and 'Between'
Both among and between date back to Old English. Among was originally a phrase (on gemonge) that meant "in a crowd," and the noun gemonge came from a verb that means "to mingle or mix." From the very beginning, among has been used to refer to a position (literal or figurative) in relation to a surrounding group of individuals, like friends, or something taken to be a composite, like a basket of flowers.

Between, meanwhile, began its life a little after among. It's direct from Old English, and is related to the word twā, which means "two." This is likely the reason some people insisted that it can only be used of two people or things—it's from the word "two"! But between originally was like among: it referred to a number of people or things. In fact, our earliest use of between in English is in reference to the Apostles, and our sources confirm that there were definitely more than two of them.

Interchangeability
That's not to say that between and among have exactly the same use or connotation, though. We use between when we want to express a relation to things and have them considered as individual and usually equal entities: between the devil and the deep blue sea; the restaurant between my house and my work; a treaty between nations. This connotation stems from an earlier use of between that referred to a point between two places, or travel between two specific points. Among, on the other hand is the best word to use when referring to things collectively and imprecisely: for this reason, among many others; no honor among thieves.

Like many rules around use, you'd be wiser to follow your own native sense of the language than hewing strictly to them. Substituting among for a more idiomatic between can create more awkwardness than it solves:

Mr. Mifflin has protested since 1914 that Ambrose Bierce would show up some day and flatly refute every single thing I have written about the amazing quadrangle that existed among Hodgson, Conrad, Nugent, and Eve.
—James Thurber, letter, 2 May 1946

Yes, Thurber used the "correct" among here, since there are more than two people being discussed. But it sounds awkward, especially given that among connotes vagueness, and following it up with the names of four distinct people, considered by Thurber to be on equal footing—in a quadrangle, even—negates among's essential vagueness.

Or consider this use of among, which was definitely chosen over between entirely due to the numbers:

The psychiatrist said under cross-examination...that he would include simultaneous intercourse among two men and a woman—a scene shown in the film—in the category of normal...
—New York Times, 30 Dec. 1972

And it continues here: https://www.merriam-webster.com/words-at-play/between-among-amongst-use

You are correct.