"Go to Game Center.
If there is a username already visible, tap where the user name is and hit the "X" to the right of it.
Touch "create new account".
Go through the prompts.
If the account is for a minor, it will not ask for an email address. A child will use their nickname along with their password to login. BE SURE TO WRITE DOWN AND KEEP IN A SAFE PLACE. Apple's Game Center support cannot view minor accounts for some reason!
If the account is for an adult, they will need their own email address.
Finish going through prompts. Now you each have your own login and password.
Now each user can go to the game they want to play. They may be prompted to select which user is playing and login.
I am not sure why some of you would have the name change on all devices. It could be an automatic synching issue when connecting to iTunes or via iCloud. Check your iTunes preferences.
Hope this will help some of you.
Also, although you each created your own username and password, even with different email addresses for login purposes, apps will still be purchased under the original iTunes account. If a user wants to make a purchase on their own individual devices, they will be required to enter the password for the iTunes account, not the password for their Game Center account.