No, not really. It was even more twisted. 16 sector was done for the Pascal System for the Apple II, independently and before the Apple III got it, but didn't 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 extremely slow, as every 7th block would be 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 controller card. It was usually also bundled with a language card (16 KiB memory card).
At that time Apple was all about getting the Apple III to market. Management believed that it should be superior in every way and no II model should have similar capabilities. Thus disk controllers were still shipped 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 second PROM onto the first and select each with a manual switch. From an Apple management point of view they were different markets (home vs. school) so they could 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 almost everyone at the company believed that the Apple II would be history half a year later. Well, history doesn't care for management and the Apple II was still runing strong, so policies got revised to tap into this by developing the LCA (Low Cost Apple) which became the Apple IIe. Its weird banking scheme is a result of the policy, still in effect at the time, that no II can be better than a III - so 128 Ki was defined as the maximum memory. But that's another story.
Long story short, to satisfy the 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.