We know commodores in TOS can (a) command a single ship and (b) command more than one ship. It is reasonable to assume that (a) and (b) are both the same officers, but they rarely act as (b) because starships normally work independently on detached patrol or exploration missions.
So a commodore is minding his own business and commanding his own vessel, when he gets a space-gram from Starfleet Command: Rendezvous at such-and-such location with starships X, Y and Z, to form task force under your command for duration of mission, and so on.
He retains command of his own ship, but to mark the transition from single-ship commodore to multi-ship commodore, he shifts from the starship badge (or whatever) to the Starbase/Starfleet Command badge. Maybe he's required to, or maybe he just has the option. But it doesn't seem too far out an explanation to me. If you want to throw the high-back chair in there along with the badge, that's fine with me; I personally don't give too much weight to furniture as a rank indicator.
Lots of people are drawing from the contemporary USN, but I think a better idea might be to look at the RN. The key here is that in the Royal Navy, commodore is not traditionally a flag rank even though it's equivalent in grade to a one-star rear admiral. You see it in the army, too, where the UK brigadier is an 0-6 field-grade officer, not a general officer.
Yes but according to Stocker in "The Deadly Years" a commodore is a flag officer.