Higher phaser types would likely mean: improved efficiency and higher yield.
No, higher phaser types mean exactly that: different
type. The higher efficiency and yield would be indicated by model number or class, and there's bound to be some overlap over the century.
You could extrapolate this even to modern weaponry. Say, for example, that a particular WW-II era light cruiser is equipped with a battery of 5-inch 38-caliber guns. You then put this cruiser in a lineup next to its modern Ticonderoga-class counterpart, with a 5-inch 54 caliber gun. In trek lingo you would refer to both guns as "Type-5 rifles," but that doesn't describe the model number or capabilities. Even weapons of the same caliber--the modern Mk-45 on the Ticonderoga and the old Mk-42s it replaced--had different capabilities and firing rates, different hitting power, fired different ammunition and had different ranges of accuracy.
The same is undoubtedly true in treknology. A Type-2 phaser of the 22nd century is probably less powerful than a Type-1 phaser of the 24th. Likewise, the presumably "Type-5" phaser cannons of NX-01 were almost certainly less accurate, less efficient and less powerful than any of the Type-5 mounts of the 23rd and 24th century, and lacked some of their modern capabilities (such as, for example, remote power beaming as demonstrated in The Cage).
By the same token, it's entirely likely that some Type-8 and Type-10 phaser banks existed even in the early 23rd century. Technical complications would have precluded their use on starships for the same reason that ICBMs have never been installed on surface warships (the phaser arrays of the TMP era were probably the size of skyscrapers). As technology reduced their size and increased efficiency, Type-8, -9 and -10 weapons became smaller and efficient enough that portable versions of them could be fielded to fleet service.
I suspect that Type XIIs are about as high as anyone would need, though who knows what they are on the NuEnt. Type XXXIV? Pew! Pew! Pew!
