seperate music libraries in family membership

Hello,

Just upgraded to a family plan on Apple Music, it says each individual gets their own music library but my music is still going onto other members phones and vise versa. How do I make a seperate playlist for each phone?

My mothers account is the one in control, I have logged onto Apple Music using her account on my phone, as logging in under my account didn’t work. Is that correct? Sorry I sorta need step by step explanation.

Thanks in advance