How can I keep my Apple Music library intact when changing iPhones?

I have never signed up for Apple music, not even a trial because I wanted to keep my music library on my devices and not rely on internet connection. Also I do not want to share all my music created by me and downloaded from rare LPs and CDs. That said, when I purchased a new iPhone 16 Pro and restored from a back up of my current iPhone 14 Pro all the music disappeared on both phones and some of my own stuff is listed in the library however it is only available as a download. How did Apple get my music and put it in the cloud. I never have used iCloud Back Up or Apple Music. Seems everything I was trying to avoid from Apple music happens anyway when you change phones. This happened to both my own music and music purchased from iTunes and online.

How can I prevent this in the future? I would like my music on my devices and never have to download again.

Anybody else have this problem?

[Re-Titled by Moderator]