'between' when there are two things, 'among' when more than two.
The second sentence is in trouble! As an adjective, we can say, the rerouted traffic takes a variety of alternate routes.
But as a verb, it can only refer to 'between two things' - The economy alternates/seems to alternate between depression and boom.
Yet the sentence, with its current punctuation, is listing three things.
Perhaps re-written as:
The author alternates between slang and clichés, and quotes from literary giants.
Actually, I'm not surprised, if you insist that "between" can only be used when there are two things. That's actually a misconception, and while the rule is usually helpful, it doesn't cover all eventualities.
We usually use "between" for reciprocal relationships (which usually entail only two things, but don't always) and "among" for collective relationships.
For example, Switzerland is between four EU member states (France, Germany, Italy and Austria); but if we were to say Switzerland were among EU states, we would be implying that Switzerland itself were an EU state, which would be incorrect.
Also, "alternate" does not necessarily refer to "between two things", although we often use it that way, but rather implies a succession of events one after the other, and can mean "rotate" or "take turns". By analogy, there is nothing to stop us from having more than one alternative.
"The author alternates between slang, cliché and quotes..." would be perfectly acceptable.
Perhaps you should tell the Cambridge Dictionary, so that they can correct it for all future users: Definition
1 [I usually + adverb or preposition] to happen or exist one after the other repeatedly:
She alternated between cheerfulness and deep despair.
2 [T usually + adverb or preposition] to make something happen or exist one after the other repeatedly:
He alternated working in the office with long tours overseas.
adjective [before noun]
1 with first one thing, then another thing, and then the first thing again:
a dessert with alternate layers of chocolate and cream
2 If something happens on alternate days, it happens every second day:
As a Moderator, perhaps you could clarify for me, whether respondents to a post are meant to respond to the specifics of what is being asked, or write a bloody treatise on the subject, particularly when they are still grasping the fundamentals of the language? Is it necessary to hit them over the head with all 24 volumes of the Oxford??
Well, I've looked through a few definitions in various dictionaries, and while most of the examples have two things alternating, none of them make it clear whether one can only alternate between two things. Some of the definitions clearly do preclude more than one thing, but others do not. The American Heritage Dictionary, for example, offers "to occur in a successive manner; to act or proceed by turns" as its primary definition without explicitly restricting this to two actions or objects (or, it must be said, allowing more than two).
In other words, it's unclear what the dictionaries are saying. This, then, would be where other native speakers cast their votes for one view or the other.
This isn't, of course, relevant to the original question per se, but you raised the subject nevertheless, and in the process recast the sentence slightly by lumping together "clichés" and "slang" (which may not have been intended by the author). Looking for a different verb that means "using more than two things one after the other and repeatedly" might have been more helpful.
And so far, I have seen no evidence that nyugaton is "grasping the fundamentals" of English; quite the opposite, in fact, and judging by previous posts and threads, I think nyugaton will be able to follow this debate.