No, not realy. It was even more twisted. 16 sector was done for the Pascal System for the Apple II, independant and before the Apple III got it, but didnt get rolled out for DOS until after the Apple III was introduced (and failed)
Wozniak developed the 16 sector format in 1979 for the Apple Pascal System, as otherwise the UCSD P-System would not only be space constrained, but also extrem slow, as every 7th block would spread across two tracks. The Pascal System went on sale as a set of disks and a PROM to be placed on the Disk II controler card. Usually also bundled with a language card (16KiB memory card).
At that time Apple was all about geting the Apple III to market. Management belived that it should be in any way superior and no II model should have similar capabilities. Thus disk controllers where still shiped with 13 sector PROMs and DOS 3.2 only supported 13 sector format. The 16 sector format was reserved for the Apple III (and Pascal). The Apple II emulation included a 13 sector RWTS, to be loaded before accessing any Apple II disks.
This created a situation that Pascal could use the same disk as DOS with about 20% more space and even worse, people using Pascal and DOS had to swap PROMs (or own two controllers). A very common hack was to solder the the second PROM onto the first and select each with a manual switch. From an Apple management point of view they where different markets (home vs. school) so they coudl have different hardware, but reality was different and users became quite demanding to upgrade while using DOS.
The Apple III was introduced in spring 1980 and close to everyone at the company belived that the Apple II would be history half a year later. Well, history doesn't care for management and the Apple II still did run strong, so policies got revised to tap this by developing the LCA (Low Cost Apple), which became the Apple IIe. Its weired banking scheme is a result of the back then still existing policy that no II can be better than a III so 128k was defined as maximum memory. But thats another story.
Long story short, to satisfy demands of Apple II users for more disk space (and to generate additional sales), DOS got a facelift and the 13 to 16 sector kit was made available.