In "The Star-Spangled Banner," "The deep" (line 9) means: the sky, the throat, leak, or the ocean?
The second stanza of Francis Scott Key's "The Star-Spangled Banner" reads:
On the shore, dimly seen through the mists of the deep,
Where the foe's haughty host in dread silence reposes,
What is that which the breeze, o'er the towering steep,
As it fitfully blows, half conceals, half discloses?
In my humble opinion, "the deep" refers to the ocean; "the mists of the deep" refers to the mist that often rises from the waters of the ocean.
Referring to the ocean as "the deep" is an example of calling a noun by an adjective that describes it; in other words, "the deep" is short for "the deep ocean." Compare this to the contemporary phrase "my bad," which is short for "my bad move" or "my bad action."
Is this something new? Not at all. Check out Shakespeare's Sonnet 18:
And every fair from fair sometime declines...
In my opinion, the first word "fair" is a substitute for "fair [beautiful] thing" (the second one is probably short for the noun "fairness"). The noun (in this case, the poet's beloved) is referred to by the adjective--"fair"--that describes it.