After a couple of weeks of searching; I've been able to find the specific reasons for all of the confusion and hopefully this might help.
Hardware
Apple only manufactures 2 versions of iPhone 4S. This is referring to hardware-only (no programming yet)
The "normal" iPhone 4S (A1387)
The China CDMA version of iPhone 4S (A1431)
Programming
During the final stages of distribution, the Modem Firmware (or "baseband" as it is often referred-to online) is programmed with specific instructions. This is what determines the capabilities of the phone and to where the phone will be shipped (either to a carrier distribution center or to Apple directly who then stocks it accordingly)
1) GSM Unlocked (available in the rest of the world as the standard, and availalbe in the US as a full-price purchase)
2) AT&T Locked
3) CDMA locked (with an option enabled on carrier-end to accept 'certain' GSM SIMs with restrictions)
There doesn't currently exist a way for consumers to "change" the Modem Firmware on their phones to switch providers via an option in the UI. So unfortunately, this means a new phone must be purchased to carrier-hop succesfully in the US. The US CDMA carriers have their own artificial barriers currently to prevent the phone from being "brought to" their network from another CMDA carrier and also disallow the GSM functionality if intended for the use with AT&T or T-Mobile in the US. Furthermore, the "unlocked" iPhone is GSM-only so it has no functionality AT ALL to talk to CDMA networks which is why it can't be "brought to" Verizon or Sprint.
Since all of this is really confusing to consumers, it is not exactly advertised, but that is essentially how it works.
Currently, there arent any phones manufactured that truly can use all of the big 4 US carriers via a UI-enabled process. And as others have said, both Sprint and Verizon disallow BYOD for Smartphones. While some of us find those restrictions rather oppressive; it is most likely done for the purposes of protecting the interest of carriers since in the US; we have the unique practice of a carrier offering subsidies (paying for your phone so you can pay less for it at purchase-time)
I have not been able to determine if the push for LTE will mean that in the futre, all of the major US carriers will use the same technology but just on differing frequencies--which may increase the chances that we'll see less restrictions like this in the future. Who knows.