Hello mediamodia,
Thanks for reaching out to Apple Support Communities. Upon review of your post, it seems that you want to change your name in the Music app on your iOS device to reflect your name change for your playlists. We can see why you’d want this taken care of, so let us help from here! Let’s start by reviewing the following article which may help: Create an Apple Music profile on Mac. Check the “Edit your profile name or photo” section. This is for a Mac, not an iOS device, but the steps are pretty much the same. You can change the name of your profile which will reflect in your playlists.
Please let us know if you have follow up questions, so we can continue assisting. Take care!