Now, it comes to a practical step. How really we can get some old cells out. It comes to the package available to us.
Today, we have lithium cells in few forms:
Also, while I attempted to write up theoretical idea behind lithium ion, someone else already did it much better. Here is video how how electricity is produced inside the battery.
SAFE than SORRY. Battery stores energy, which transfer to heat and fire. The cell can release nasty smell similar to alcohol or ester.
This is the form you can access easily since we have at least one laptop around. Laptop battery is wrapped by clampshell plastic. Two plastic pieces are locked together by inside dovetails. You need a flat-head screwdriver and go between the locking. Notice that don't prick or punt a sharp metal directly into the cell. Go around and unlock dovetails.
Polymer lithium battery (Lipo) is found in newer models. Those battery comes in a flat shape rather a long and removable pack.
This type can be excellent since the pack is replaced when the battery went LOW, not just die as other electronics.
You may notice that I leave extra tin taps attached to the cells. There is a good reason for it. If you have a spot welder (most of us don't), you will be fine to solder them, but if you have a tin solder, making a good joint to a terminal is not easy. In fact, trying the solder tin can heat up termials and decease your battery capacity. You can eventually destroy the cells if pushing the hot iron to the cell for a good joint. That extra tin taps make easier to re-wire them later, such as building back a 3S (3 cells in parallel) for solar charger.