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I have seen other topics discuss this, but I stumbled across some information that helps clarify the rank of command aboard the Death Star. This is from the official screenplay of Star Wars A New Hope (4th draft 1976)

Suddenly all heads turn as Commander Tagge's speech is cut
short and the Grand Moff Tarkin, governor of the Imperial
outland regions, enters. He is followed by his powerful ally,
The Sith Lord, Darth Vader. All of the generals stand and bow
before the thin, evil-looking governor as he takes his place
at the head of the table. The Dark Lord stands behind him.

We see this indicates Tarkin is the governor of the Imperial outland regions (which I suppose includes Alderaan and Dantooine, since he is onboard as they approach both planets). This also shows the term "The Sith Lord.." was mentioned in the screenplay and the book by GL if not spoken in the dialogue.

Later in the conference on the Death Star:

The bitter Admiral Motti twists nervously in his chair.
MOTTI: Dangerous to your starfleet, Commander, not to this battle
station!

This reference plus another where Tarkin orders Motti to proceed with the attack on Alderaan would indicate Admiral Motti is actually in charge of the Death Star itself. So Tarkin is on board as the Imperial Governor of the region the Death Star is patrolling as they search for the Rebel Base, Vader is onboard following Leia since he's after the Death Star plans, and Admiral Motti has to take crap from both of them because they outrank him. But when they aren't around he's top dog.

True, but Vader was gonna kill him, after he badmouthed the force, meaning he was expendable, whereas someone like Tarkin was not.

The term commander is vague. Luke was "Commander Skywalker" but there was no specification what he was commander of. There's no real telling what Motti was in command of, but given that that meeting with Tarkin and Vader, it's safe to assume that everyone in there was commander of something.

I think Luke was commander of Rogue Sqaudron, wasn't a commander of anything else.

Motti was in charge of the Death Star. Tarkin and Vader were supervising the entire project as it was the Emperor's super-plans and they were there as his representatives (and there's no reason to think that Tarkin would not have been there if it was not in the outer regions- you don't have to stay where your governorship is; he's just a super-powerful Imperial politician). Vader did the heavy lifting and Tarkin called the shots.

Indeed the script mentions the Sith and the film does not. GL did not write the novel, however.

Well said ush

I guess you could make the argument that Tarkin was the main villain of ANH and Vader let him do the planning and such.

Tarkin was the real villain in the first movie

Everybody remembered Vader as the villain, he made more impact, even being on a leash.