Doing a search against use of "levels" in TOS it would appear that the Engineering deck possesses multiple levels. Levels do not appear to be referred to on other decks so it might be safe to say in "The Naked Time" and "Omega Glory" that "level two" and "lower levels" were understood to be on the "Engineering deck". Interestingly, then if the engineering deck starts at the Engine Room there would be a maze of "levels" at the top (think EMM) going into the bowels of the engineering hull...
The Enemy Within: SCOTT [OC]: Mister Scott, sir, on the lower level of the Engineering deck.
The Ultimate Computer: SPOCK: There is one possibility. The automatic helm navigation circuit relays might be disrupted from engineering level three.
The Naked Time: UHURA: Sir, level two, corridor three reports a disturbance. Mister Sulu chasing crewmen with a sword.
The Omega Glory: SPOCK: Lieutenant Galloway and I are checking out the lower levels. There seems to be no one aboard. Only uniforms.
The Man Trap: SPOCK: Supplies of salt have been set out as bait at all decks and engineering levels, Captain. However, no one or nothing has approached them as yet.