Actually, Ivan is correct.
The Applecare warranty starts as soon as you register it.
You see, without it, phone support is free for only 3 months.
With Applecare, you get free phone support for 2 years.
So saying Applecare starts after the factory warranty expires, to me is incorrect.
In fact, it clearly states this from the link that you provided.
Well actually, we are all correct (sorry Ivan I stand corrected). Though I think we are splitting definitions a bit here. The context of the question was within the subject of the warranty coverage within Applecare. The link clearly states that the Applecare portion of hardware (Repair or replacement) coverage begins after the included 1 year warranty. However, you are correct with regards to the phone support beginning the moment you activate Applecare. I interpreted the context of the OPs question was only in regards to hardware warranty, however I may have incorrectly comprehended their question.
With that said, I still stand behind my earlier school of thought (not sure if that sentence works). If you do not need to use phone support (or otherwise) within the first year, it makes total sense to postpone buying Applecare until the 11th hour, or not at all. Also, if you did happen to buy Applecare on your purchase date, and decided not to activate it, you will still be able to sell your unused Applecare, or simply apply it to your next iPod that is compatible with that warranty. A good example would be an experience my friend had. He bought the first generation iPod touch 16gb and Applecare. He never activated Applecare on the iPod, sold his iPod Touch, and got an iPhone 3G. Still having Applecare, he then sold the unopened (unused) Applecare protection plan to a friend for what he purchased it for. While you may not be able to get all your money back in every example, it is still not a bad option to consider.
So to me, we are both correct, but I like the idea of waiting to activate (or purchase) Applecare until the 11th hour, especially if you never need phone support within the 1st year, or outside of the first 90 days.