A rather truer explanation of the limitations of the 'rule' is to be found at Textbroker:
Commas and Coordinating Conjunctions (FANBOYS)
You've seen the commentary, but what does that mean? Coordinating
conjunctions are all of the following:
For .... And .... Nor .... But .... Or .... Yet .... So
You can remember them by being FANBOYS of good grammar. If you're
having trouble with comma usage each time one of these pops up, here's
a trick that can make it easier. You only need a comma when each
part can stand on its own. So whenever you see one of the FANBOYS,
split the sentence around it.
I went to the store |and| bought eggs.
"I went to the store" is a complete sentence, but "bought eggs" is
not. This sentence is fine the way it is.
I went to the store |and| I bought eggs.
"I went to the store" is a complete sentence. "I bought eggs" is also
a complete sentence. The sentence should read: I went to the store,
and I bought eggs.
You can do this with any of the FANBOYS.
I went to the store, |but| the chickens were on strike, |so| there
were no eggs.
I went to the store |but| found no eggs.
They say women are from Venus, |yet| Mars already has Martians, |so|
why do they need men too?
If you can split a sentence, you can join the full-fledged FANBOYS.
With the trend towards light punctuation nowadays, I'm sure even this attenuated 'rule' is over-prescriptive. I'd be perfectly happy with
I went to the store and I bought some eggs.
I'd insert the comma where it would sound decidedly off not to insert a pause when reading aloud.
Oh, and the FANBOYS analysis has come under heavy criticism, as this article by Jack English (?)describes.