Amusingly this is somewhat game-specific to FFXIII and FFXIII-2, and the source I referenced explains why (Knurek on NeoGAF), though you can test it yourself in a variety of ways including the utilities on PCGW:
"The fact that FFXIII team wrote it's own XInput implementation, instead of using readily available xinput.dll, really shows just how much SEJapan doesn't get PC developing."
Knurek is saying that instead of using the XInput DLL or XInput's driver interface, SEJ rolled their own "XInput implementation" (in this case using DirectInput) according to the documented button layout for Xbox controllers without realizing they didn't actually implement XInput at all because:
1) DirectInput uses the controller's HID driver path (outside of hooks it's similar to RawInput's HID interface in XP+), while XInput runs on a separate driver path (the Xbox 360 controller drivers supplied by Microsoft don't implement vibration or the LT+RT buttons in the HID path, for example).
2) Various third party programs rely on XInput DLL hooks to provide alternative behavior.
You can still use XInput without the high level interface in xinputXX.dll, though nobody in their right mind does so because of #2 and the pointless added development time.
Then again, this appears to be flying past everyone's heads here (it's not complicated at all to me, but like I mentioned I have a decent amount of wingamedev knowledge which appears to be making it difficult to explain in a simplified manner), so I'll still leave it as is and just make a correction to your version.