The Lenovo warranty world is both confusing and unbelievably comprehensive.
There are multiple types of every warranty, as you have noted, but many more than you've noted. I won't try to explain them all, but I'll list my three greatest warranty tips.
1) YOU CAN ONLY UPGRADE A WARRANTY ONE TIME on a machine that is still covered under it's original warranty.
Example: If you buy a T520 with a one year depot warranty and you upgrade the warranty to a three year depot warranty (very typical upgrade), then your done. You can no longer upgrade to on-site, TPP or five year depot, for instance. You are allowed only ONE upgrade while during the original term warranty that was attached to the model number of the machine. Using the example above, even though the T520 is eligible for a five year depot warranty, if you've purchased the three year upgrade, you can not upgrade to five, even if you try before the three year expires. ONE upgrade, that's it. However, you can buy a POST-warranty coverage! It will cost a lot more, but you can add two more years that way.
This is why it's important to look up the warranty yourself to verify if the machine is still under warranty and if it's been upgraded yet.
2) YOU CAN UPGRADE MOST THINKPADS TO FIVE YEARS OF COVERAGE! This is great, since many machines are fine (in some capacity) for five years or longer. (I'm currently using a 2006 Z61m for work.)
3) THERE ARE TWO TYPES OF THINKPAD PROTECTION PACKAGES/PLANS (TPP) - BASIC AND REGULAR! It is very important to know which one you are buying since they are very different and they are priced differently. Basic provides for a SINGLE repair claim over the life of the machine! Regular (actually not called that, but just TPP vs. TPP Basic) provides for UNLIMITED repairs of the life of the machine UP TO THE COST OF THE SYSTEM. Neither is that great, but regular is better.
T520, i5-2520m, Optimus, HD+, 8GB, Sandisk Extreme SSD, N, Webcam, BT Home
T400s, C2D SP9400, WXGA+, 8GB, Intel X18-M G2 SSD, AT&T WWAN, N, Webcam, BT Travel
Edge 14 Core i3| Edge 15 Core i3 | Edge 15 Athlon II| Edge 15 Turion II