Definition of "dependent clause"

In

"He said that the laptop is shut off," "He said that the man is strong" [...] "He said the sky is blue."

everything after "he said..." is a noun clause. The Danish Grammarian Otto Jesperson called these "content clauses". In your example, they are all the object of the verb "said"

They are not subordinate to anything any more than in "He hit the man", "man" is subordinate to anything.

Wikipedia has an article on content clauses

In English, there are two main kinds of content clauses: declarative content clauses (or that-clauses), which correspond to declarative sentences, and interrogative content clauses, which correspond to interrogative sentences.

Declarative content clauses can have a number of different grammatical roles. They often serve as direct objects of verbs of reporting, cognition, perception, and so on.

The definitions given are misleading.

Try the Oxford Dictionary of English Grammar p399 definition of subordinate clause

subordinate (adj.) (Pronounced /səˈbɔːdɪnət/.) Grammatically *dependent. Contrasted variously with *coordinate, *independent, and *superordinate.

subordinate clause: a *clause that is dependent on, or forms part of, another clause, *phrase, or sentence element. Subordinate clauses can be classified in different ways, e.g. as being *finite, *non-finite, or *verbless, e.g.

They think that the swimming pool is open today (finite **that*-clause)

They want to eat pizza tonight (non-finite **to*-infinitive clause)

With you here, things will be easier (verbless)

Traditional grammar recognized three types of subordinate clause:

(i) *adverbial clauses, e.g. I was surprised, because it was so unexpected

(ii) *nominal clauses, e.g. It was odd that he didn’t telephone

This includes *nominal relative clauses, e.g. I was surprised by what you said

(iii) *relative clauses, e.g. The news (that) you gave us is very odd

In more recent frameworks (e.g. CaGEL, OMEG) three kinds of finite subordinate clause are recognized, namely *content clauses, relative clauses, and comparative clauses. Non-finite clauses are classified into **to-infinitive clauses, bare infinitive clauses, -ed clauses, and -ing clauses.

Another kind of clause that is sometimes classified as subordinate is the *comment clause, e.g.

We will, I think, see a lot more of him soon

The above manages to define the concept just fine without recourse to such nebulous concepts as 'complete sentence', 'complete thought', and also has no mention of a subject being required.

Also, p115 dependent element

dependent element, phrase, clause, etc.: any unit other than the *head within a *phrase, e.g. a *modifying phrase inside another phrase (the dead rat); a subordinate clause functioning as *complement of a verb (I think that it’s too late for coffee), etc.