I think that you are confusing two different situations.
Without more details, I strongly suspect that the "Great Writing" series, Grammarly, and Google search are telling you not to use a comma between two conjuncts when they are the only conjuncts in a series. For example: "I have two cats: Fluffy, Mittens." The illustration that you included from Grammar Monster says the same thing. (Note that this rule is not absolute and that occasionally a comma actually works well in such situations.)
The Saki story, on the other hand, uses a coordinating conunction to separate two conjuncts: "a horse struggling with a more than ample load" and "a carter of the sort that seems . . .". However, Saki has decided to make the second of those "parenthetical" (or "nonessential", etc.). He has therefore surrounded it (along with the preceding conjunction) with paired commas. The second comma would appear at the end of the sentence (immediately before the period) and has therefore been omitted. What he's done is entirely acceptable and quite common. (Personally, I see no reason to make the second conjunct parenthetical and therefore would probably have omitted that comma, but that is to a certain extent a matter of opinion.)
Note that some people would advocate for the comma on grounds of clarity, arguing that it helps to distinguish the conjuncts in a rather lengthy sentence.
TLDR: The grammar sources that you cite discuss use of the comma to separate conjuncts, while Saki uses the conjunction "and" to separate conjuncts. He uses the comma for an entirely different purpose, so the grammar sources' advice doesn't apply.