Donny, I'm glad to have been of service to do my part to ensure that fact doesn't become myth (because of deteriorating reference material) and that myth (...) doesn't become fact.
Although I'm really not trying to find fault and be more nitpicky than necessary, please note the asymmetry of the admiral's braids:
The braid below
the Commodore braid is only separated from the Commodore's braid by a space equalling the width of one
braid. In contrast the braid above
the Commodore braid appears to be separated by the width of two
I also noticed that you didn't consider "Fleet Captain" (e.g. Pike) as an in-between rank. Admittedly, we never saw such rank but if you wanted to bring the "lower" ranks to "full circle" I presume this rank would consist of three uninterrupted braids.
If I remember correctly, starting with "Commodore" we are on the flag officer level. At this level interrupted half-braids are somewhat inappropriate, I believe and therefore we do only see full braids in addition to the Commodore braid.
There is debate among fans whether Fitzpatrick (the only one where we saw an admiral's braid) is just an "admiral" or in fact the "Fleet Admiral" (the existence of this rank was established by a remark from Dr. McCoy in "Space Seed").
Now, just as there wasn't any real need to explicitly state "Engineering Deck 12" in either "Mudd's Women" or "The Enemy Within" (everyone knows that "Deck 12" cannot be a saucer deck and therefore has to be one in the engineering hull), I think it's not too farfetched to assume that Kirk understood that the fleet admiral was giving him a call when Uhura said "Admiral Fitzpatrick" and not just his usual sector commander (I'd expect that Nilz Baris would not have accepted just to talk to Kirk's sector commander, but insisted to speak to the fleet admiral himself. He was such a kind of guy...).
Therefore, I believe that the average "admiral" just wears one full braid above the Commodore braid while Fitzpatrick's extra braid below the Commodore braid identifies him as "fleet admiral".
Of course, this is entirely conjectural but I wanted to mention this option.