You understand the difference quite well. We consider the conjunction to be a part of the clause that it introduces.
As you write, the coordinating conjunctions introduce independent clauses. As such, they can be punctuated as independent sentences. (Some teachers dispute this position, but they are wrong.1, 2) Thus, And I teach history is an independent clause.
The subordinating conjunctions and relative pronouns introduce dependent clauses. Such clauses cannot be punctuated as independent sentences. Doing so creates a sentence fragment. Creative writers often write fragments deliberately, but fragments should be avoided in formal writing.
With reference to your specific question, because I was extremely tired should not be punctuated as an independent sentence. It should be attached to an independent clause.
As to your other terms, traditional textbooks call a sentence compound when it contains independent clauses joined by a coordinating conjunction. They call a sentence complex when it contains an independent clause and a dependent clause. They call a sentence compound-complex when it contains independent clauses joined by a coordinating conjunction AND a dependent clause. These labels are pretty useless.
1 and. A. Beginning Sentences with. It is rank superstition that this coordinating conjunction cannot properly begin a sentence.
2 but. A. Beginning Sentences with. It is a gross canard
that beginning a sentence with but is stylistically slipshod. In fact, doing so is highly desirable in any number of contexts
Bryan Garner, Garner's Modern English Usage