There is no one correct way, as the battery was not designed to be replaced in the first place. IMO, it could not have been expected that the computers would be used this long. If you want to make it like the original, you can still get those batteries at electronics suppliers. You could remove the old one, by cutting and desoldering, and then solder in the new one. It should last for another several years. The lithium coin batteries are indeed non rechargable, but that's fine. You rarely need to replace them, and they don't leak. You also need to install a diode to stop reverse current going through the battery (this is what would otherwise charge a rechargeable battery). Schottky (barrier) diodes are good for this, as they have low forward bias drops (less of the battery's voltage drops across the diode).