I had today off work, so I was able to do a little more research. I called Apple this morning, and unlike the previous two times I did so, I was *not* told to go to the Apple store before the 14th to add AppleCare+. This representative told me that I would need to take my sealed phone into the Apple store on or after the 14th to order it.
I then called Verizon again, since I ordered the phone through them (just to be thorough, I knew they wouldn't have any ideas). The gentleman I spoke to there was very nice, but they don't even have AppleCare+ in their system yet, and he said that working with Apple is different from working with other phone providers (like Samsung and Motorola, I guess) in that Apple hasn't shared any information with them. This particular man didn't even know the 4S was launching until the morning it happened when he "saw it on the news." This confirmed my opinion that Apple was the place to go.
I had to go to Manhattan today anyway, so I stopped by two Apple stores to test out the "bring the confirmation email in" advice that I was given in previous calls to AppleCare. The store on 14th Street said they had never heard of such a thing, "absolutely not." That representative was not very helpful, and was not as friendly as the other people at Apple that I have talked to. I then went to the flagship store on Fifth Avenue, thinking if any place would have information, it would be that location. There I was told that they couldn't give me the AppleCare+ right now, as they didn't even have a box for it in the store. However, they were aware of the glitch on the website saying it had to be ordered at the time of purchase. They told me that AppleCare+ will be available on the 14th, as we all know, but that I do not need to come into the store on that day to activate it - I should be able to purchase it on the website as soon as I have my 4S's serial number, or I can purchase it within the first year I have the device, in person or on the site.
Since this ties in with Apple's general policy on regular AppleCare, it does make sense to me. I hope I can add it when my 4S comes on Friday!