From the page you linked, the very first note. Click that link. Here it is:
http://smile.amazon.com/gp/help/customer/display.html?nodeId...
The first line is very important: "Items sold by Amazon.com that are marked on the product page and at checkout."
Emphasis mine. As an example, I ordered a Logitech Mouse. The item was listed as Prime. It was shown in stock. It showed a guaranteed delivery date within 2 days in the item page. I simply ordered and didn't pay attention to anything else.
It showed up a week later. When I called, it was pointed out to me that the "guaranteed delivery date" that matters is NOT on the item page. It's on the checkout page and those emails they send you after you order. They were correct, the email I got said a week later ... of course, I didn't pay attention, because most things come in 2 days and the item page did say it'll come within 2 days time before I ordered.
Is that false advertising? Sure. I forgive them.
I've also ordered Prime items that were listed in stock and had a guaranteed delivery date on the item page, but then mysteriously went out of stock later and simply weren't shipped at all.
I order a lot. Maybe 2-3 items every week. This has happened only a few times. With the breadth of their operations, these hiccups are understandable. I did manage to get a few 10 dollar credits once in a while, but sometimes they just refuse to own up to it and tell you to get lost.