Semicolon. It was the best of times; it was the worst of times.

[milo]

If you don't like my #4, you could always change your #1 from:
"1. to join 2 independent connected clauses without a conjunction."
to:
"1. to join 2 independent connected clauses with or without a conjunction."
which would then include the conjunction "as a matter of fact"
and obviate the need for #4.

You are mistaking it for # 2. Try removing the comma and see if it still works . . .

It's illogical to include some conjunctions and exclude others;
so, I guess, we're back to having a #4.

It is covered by # 2. It is actually the only thing that is covered by # 2.