What you describe in your opening post is not a purchase. You are confused about the difference between purchase and rental / leasing (with the option to own).
when i sign an agreement to pay for something and after so many payments i stop paying & keep it.... that is not a lease.
AT&T does not offer a leasing program where you're able to just pay and pay and then turn it in....you sign an agreement to pay full retail over the course of a time frame