First off, let me explain a battery is one or more cells.
There is no such thing as a proprietary cell, only proprietary batteries. A manufacturer matches some cells and puts them in a case. Laptop manufacturers don't do that themselves; they buy them from battery manufacturers. ..
Absolutely correct.
Inside of the battery pack for your laptop are cells. These will look a lot like the AA or C or D cells or the CR123 cells that you put in flashlights... except that they probably have wires coming out of them instead of point contacts because they're intended to be wired in.
Inside of them, these cells are made of cadmium, chromium, lithium, nickel, lead, mercury... not exactly sugar, spice, and everything nice. These metals are very reactive which is why they can store electrical energy as chemical energy and then react again and change chemical energy back electrical energy. That same reactiveness makes them poisonous (they react with chemicals in your body in bad ways), environmentally destructive (they react with chemicals in the environment in ways nature didn't intend), and actually dangerous (they can react in energetic, explosive, or inflamatory ways). Not nice stuff.
Not surprisingly, making cells is not something that most folks want to do. And it's not something which can be economically done in the "first world." Nope. Battery cells are mostly made in China and India and "third world" countries. (We could discuss what our cordless, rechargable lifestyle is doing to the people and environment of these countries, but that's more of an issue for the Political forum.)
Also, not surprisingly, the manufacturers of these cells fall into two categories: first, there are a few major makers who are well-known and try to attract the business of the major OEMs with high-quality, well-made products. And, second, there are quite a few grey/black-market, no-name manufacturers, shady companies who come and go, who chase the cost-driven and secondary markets.