If you are reading the sentences, the quotation marks make it clear that the words are an actual quote from John. However, if you are listening instead of reading, the word “that” implies a paraphrase of what John said, rather than a direct quote.
ETA: Skimmed the OP. Both examples in the OP are OK with “that” included because they are a logical continuation of the rest of the sentence. However, “that” could be omitted in the first example (with a comma after “noted”) but not the second.
But if your style guide says you can’t keep the “that,” you can just “which” instead. Unless your style guide is one of those stupid ones that says that which can only be used for nonrestrictive clauses.
First of all, I don’t think that a style that maintains a distinction between which and that is fairly characterized as “stupid.” It’s a style preference.
Second, I think it’s doubtful that cases in which “that” can be used or not will be amenable to “which” substitution.
I had an English teacher who insisted that you should never use “so” in an essay unless it was immediately followed by “that”. You can put all your superfluous "that"s there!