Is the distinction between "complex sentences" and "compound sentences" real? Is the distinction purely a matter of personal taste?
Solution 1:
Coordinating conjunctions are not limited to joining clauses. At least, most of them aren't. Examples like "not rain nor sleet", "strange but true", "fact or fiction", "odd yet endearing" and "tested and approved" help to show that these conjunctions create coordination of like constituents. Those like constituents can be, but need not be, independent clauses.
"For" and "so" are included in many lists of coordinating conjunctions. I suspect that those lists are simply wrong. Of course, if you're trying to land a tutoring position, you might be stuck with using the wrong list for the sake of your students passing set tests.
Your stipulations for a "binary sentence" have nothing to do with clauses and their relationships. For what it's worth, this sentence is complex. For the sake of another example, this sentence is neither complex nor compound. If you try to identify those through patterns of commas and conjunctions alone, you will fail.
Your answer 1 of 3 is reasonable, if we assume FANBOYS is a good list.
Your answer 2 of 3 is reasonable, if we assume that "for" is a function word that takes an argument.
Your answer 3 of 3 is something else. It claims something that just doesn't make sense.
She did not cheat on the test, for
That looks nothing like a dependent clause. That looks like an independent clause, followed by a preposition in desperate need of an argument. That poor, stranded "for" is all alone, without any kind of object or complement in sight.
Perhaps dependent clauses seem incomplete. For many people, that's a reasonable description that works rather well. That does not, however, work in reverse. Not everything that seems incomplete is a dependent clause.