How can I log into Apple Music with a different Apple ID?

I have two Apple IDs. My phone is logged into my iCloud.com Apple ID, but my Apple Music is set up under my other Apple ID. How can I log into Apple Music with my other Apple ID? I have tried deleting the Apple music app from my phone and reinstalling it to try and login under the other Apple ID, but it constantly defaults to my iCloud.com Apple ID that my iPhone is set up under. Even when it prompts me to change it in the app via a web login, i can change the Apple ID to the current one on the web, but it still doesn’t change the ID I’m logged into under Apple Music on my phone. Please help! All of my playlists and owned music are under that other Apple ID.

[Re-Titled by Moderator]