Music library

So i have a 4 month payed music app.So i ve downloaded some playlists to see in library, knowing that if i do that i can listen without consuming internet data. Corect me if i am wrong, i am new to iphone, and starting to hate it.

So as you can see in the printscreens , i ve downloaded 5 playlists, but none of them appear in library, they appear in library only when i am downloading smt else. And dissapear after a while.

So in picture 1 you can see it was downloaded.

In picture 2 , you can see that i have more music in library , but i can see that music only while i am downloading smt else.

And in picture 3 , after i close app and reopen i have nothing else left.

But if i search a playlist again, it appears that i already have it, i can t add it again. I can delete it from library even though it doesn t appear.