That's correct. In Menagerie, McCoy placed Spock under arrest under his authority as "senior officer present," but he was not functioning as acting captain. That might seem like a hair splitting distinction to some, but it is how a ship would tend to operate. In fact, it seems that Lieutenant Hansen (navigator) became the acting commander, presumably (although this is never stated on screen) because Scotty (the official third in command) would have been occupied trying to undo Spock's sabotage.
