You're putting too much value in those terms. Don't get me wrong, what you think it should say is probably how it is supposed to be, but those songs are yours forever, as long as you have them on a hard disk or other storage medium. The kind doesn't change that fact, unless there is DRM on them. Purchased songs are DRM free, so you're safe.
As for Apple Music AAC, they're not yours, they're from Apple and you have the freedom to listen to them as long as you're an Apple Music subscriber. From the second you stop being a subscriber, Apple will delete them from iTunes, which is their prerogative.