I am wondering if I am missing something about specialties with weapon types. Is there any benefit to listing off a specific weapon, rather than the category of the weapon? On page 309 of the main book, Octavian is listed with Melee 5 (Equitable Resolution +3). A few pages later, Tepet Lisara is listed with Melee 3 (Swords +2), and they list her Jade Reaver Daiklaive named Flame's Kiss. This bouncing back and forth between extremely specific specialties and more generic ones happens quite often across several of the source books. This may just be for flavour text, but would there be a benefit of specifying Flame's Kiss instead of Daiklaives or swords?
Exalted isn't just about mechanics
The mechanical effect of choosing a narrow specialty over a broad one is that you get to apply the specialty less often. On a purely stats-focused view, this is silly, as it reduces the character's power.
However, Exalted is by no means a stats-focused game. A player chooses a narrower specialty because it tells you something about the character. This weapon is important to them. So important that they have trained with it to the exclusion of all other weapons of its type. Now we know something about how this character should act, how they might react to the loss of their favorite weapon, etc. This is much more valuable in a story-focused game than a couple more dice into the generic die pool.
And one more thing
Unlike most roleplaying games, Exalted doesn't carry an expectation that characters will level away from their starting weapons. If I have my trusty Daiklave Splitter of Mosquitoes at Essence 3, well, I'm probably still going to be using it at Essence 8, because weapons don't really vary that much in power level. So there really isn't any wasted investment in specializing to a particular weapon as an Exalt, because you're going to be fighting with it exclusively probably your whole life.