You're not doing that either. You're buying a revocable license to access. That's actually worse than renting or leasing.

In the case of Amazon Music, if you haven't fallen for the subscription nonsense, you actually are buying the MP3 files. You can download them and then it's on you to move them around as you need. Amazon will also "keep a copy in the cloud for you, and let you play it wherever." You lose that when you lose your account, but if you went to the trouble to download your MP3, that's yours.