It’s actually the norm in English grammar—not a matter of whether it’s better terminology or not—to use the terms “reporting clause” and “reported clause” for the two clause components of this sentence that I presented in my previous posting: “The manager said that workers who do highly repetitive routines need more frequent rest breaks.” As you correctly pointed out, the main clause (“the manager said”) is the reporting clause, and the subordinate clause (“that workers who do highly repetitive routines need more frequent rest breaks”) is the reported clause. Sentences of this form, whether the reported clause is a quoted statement (direct speech) or a paraphrase of that statement (indirect speech), are all categorized in English as reported speech.
The common element in all reported speech is, of course, the reporting verb, and “say” is just one—but the most often used—among the wide repertoire of reporting verbs in English.
Other than “say,” the reporting verbs that are typically followed by a “that”-clause are the following: “add,” “admit,” “agree,” “announce,” “answer,” “argue,” “boast,” “claim,” “comment,” “complain,” “confirm,” “consider,” “deny,” “doubt,” “estimate,” “explain,” “fear,” “feel,” “insist,” “mention,” “observe,” “persuade,” “propose,” “remark,” “remember,” “repeat,” “reply,” “report,” “reveal,” “state,” “suggest,” “suppose,” “tell,” “think,” “understand,” and “warn.”
Some verbs can be followed either by a “that”-clause or a “to”-infinitive, namely “decide,” “expect,” “guarantee,” “hope,” “promise,” “swear,” and “threaten,” as in the variant sentences “The committee decided that the anti-corruption hearings be continued” and “The committee decided to continue the anti-corruption hearings.” A few other forms of clauses can follow certain reporting verbs, like the “if”-clause in “They asked if we are joining the picnic” and the “whether”-clause in “See whether our farmhouse is still flooded.”
Whatever form of clause follows the reporting verb, however, the defining characteristic of reported speech is that what’s written or being spoken about is what somebody else has said, thinks, or believes.