I could be way off the mark here, but I’ve never seen this as a hard or fast rule that the plain past is only used to report what the speaker has said.
Rather, there are a few elements that influence which form is chosen.
First, 言った deals with a single utterance, right? If the speaker was talking at length about some topic(s), then 言っていた is more appropriate, similarly, if something was repeated multiple times or said habitually, then you’d still use the continuous past. Essentially, any plurality of utterances would utilize that form.
“He said X” vs “He was saying X and Y” or “He was saying things like that”.
“Dad used to always say that.” “He often said those kind of things, but… (he didn’t mean it)”
Secondly, the speaker’s certainty (or lack thereof) of the validity of the statement has an influence on which form to use, and I think this is essentially what Misa is trying to get at in her explanation.
と言った。 is very matter of fact. If the speaker is relaying something that was just said involving a state of affairs (Tom said he is busy) or an opinion (Susy said she likes cats) it is safe to use と言った since we can assume that the validity of this statement has not changed.
If enough time has passed where the speaker can no longer be certain, then it becomes more appropriate for the speaker to use と言っていた because there is no way for them or the listener to know whether what the other person said is still true. “Sam had said that he wanted to go to China (but I don’t know if they still do…)”
“Tom had said he was busy… (but that was hours ago, maybe things have changed)”