DirectX 9 was mainly intended for use on Windows Vista and windows 7 machines, and the technology available at the time.
When those operating systems were released, you often saw "Gaming PC's" with dual-core processors, 8GB of RAM, 1GB of VRAM, and so on. DirecX9 was specifically catered to look best, while performing the best on these generation of computers.
DirectX 11 was released instead on the next two generations of Operating Systems, Windows 8, and Windows 10.
Windows 8 was a nightmare to check email on, let alone try to run a game such as Crysis.
Windows 10 was, however, done a massively amount better, in nearly all aspects.
The Windows 8 and Windows 10 generation of PC's had specs, often including Quad-Core processors, 16GB of RAM, and 4GB of VRAM.
DirectX 11 was catered to these new specifications.
When your computer is lacking in one or more of the critical areas, DX11 will often fail your expectations.
TL;DR, DX11 has great quality images, but DX9 is built to be less power-hungry.
Now, as for the i5. I have a few machines with i5 quad core processors in them. I also have a few with i7's.
The i5 was great for its' time, however, intel never did nail down the mutli-threading on multiple cores on the i5 like they did with the i7, and they're still screwing in a few loose nuts and bolts.
Often, either system services, the CPU, or the device itself will cause this input lag - some game settings may as well.
SUPER TL;DR, Your PC isn't up to par to run DX11 smoothly. I suggest you sacrifice some of the glorious graphics, and tone back down to DX9.