After I switched out the cartridges, I shipped the empty cartridge back in the same box as I got the new cartridge, print out a shipping label and drop it off at the post office. Hence, I "rented" the cartridge and kept the ink.
You're not renting the cartridge; it belongs to you. You may or may not get a credit towards the purchase of a new cartridge if you return your old one for recycling, but there is no penalty for simply keeping it. If you were renting the cartridge you would be obligated to return it eventually, whether or not you wanted a new one.
There are cases where the container for a consumable really is rented; for example, if you need a small quantity of liquid nitrogen you'll generally want to rent a dewar to carry it rather than buying your own. (Liquid nitrogen is relatively cheap, on its own, but the dewars start at several hundred dollars.)