What part of 'Music' do I move to external hard drive?

Getting totally confused with Apple Music and my personal music library. I want to be sure I can use Music app on my other devices and if necessary burn music or share music. If I put music on an external hard drive, do I need to actually move the music folders (by artist) or just the iTunes library please? I tried moving the complete Music folder which contains all music and Apple Music and iTunes and changed the preferences to show as much but the track details still show that the file is on my Mac HD? 😵‍💫