The article you found—where both arguments and explanations are specific types of rationales—may be fine in a narrow use case, but English is broader than that.
As a quick example, here are three senses of the definition of argument:
1 c : an angry quarrel or disagreement · having an argument over/about money · trying to settle an argument
5 a mathematics : one of the independent variables upon whose value that of a function depends
b grammar : a substantive (such as the direct object of a transitive verb) that is required by a predicate in grammar
In the case of a philosophical debate, somebody could easily use the word argument, meaning something that is a type of rationale. But when gossiping about the latest celebrity arguments, people aren't going to assume that what's being discussed is a reasoned thesis and antitheses; instead, they will assume that it's two emotional people shouting at each other.
Because of the associations people have with word meaning in different settings and contexts, it might make more sense for somebody in a casual setting to say "What's your rationale for saying that?" than it would for them to say "Give me an argument for that."
Beyond that, certain phrases are simply more idiomatic—they are more commonly used.
For instance, I suspect most people in a casual setting wouldn't say either "What's your rationale for saying that?" or "Give me an argument for that." Instead, it's much more likely they'd simply say, "Why do you think that?" (Or "Explain yourself.")
But by the same token, in some contexts, rationale will simply be more commonly used than either argument or explanation.
Another situation in which rationale might be more suitable is one of ease of use. Assuming the validity of the analysis that an argument is one type of rationale and an explanation is a different type, saying rationale in every case saves you the trouble of figuring out which of the other two words is the specific one to use.
In such a situation, you might use either argument or explanation incorrectly—but (or so I'll argue here) you can never go wrong by using rationale.
(Of course, the reason for not using rationale could be that you want to be precise.)