- たくさんの階段を昇った後だったので彼女は完全に息を切らしていた。1 (original example sent.)
My guess: there is a strong causal link between this and that in 1. 2 just means "A happened, then B happened". Is that correct?
Exactly as you say.
"After running up so many flights of steps, she was completely out of breath. "
The English translation from that page is not topnotch IMHO because it fails to reflect the strong causal link that you speak of. More specifically, it does not even translate the 「だったので」 part.
That looks more like the translation of your second Japanese sentence.
My own literal-as-possible TL of the first sentence:
"Because it was (right) after she ran up many flights of steps, she was completely out of breath."
That should show the cause-and-effect relationship between the clauses as clearly as the original sentence does.