I find it interesting that Vader is referred to as Tarkin’s “henchman.”
You certainly sense in the movie that Tarkin has some authority to restrain Vader, but hardly that he is in charge.
“I should have expected to find you here holding Vader’s leash,” Leia says. I mean I think it’s pretty clear that Tarkin was the man in charge and Vader was just sort of the muscle.
That’s another thing retconned in the prequels, making Vader into some kind of god figure (complete with virgin birth) and centering the entire saga on him. He wasn’t even originally supposed to be second in command, at least not officially. Even in Empire, we see him storming Hoth on foot at the front lines. He’s the muscle, not the brass.