I obviously clearly explained on how to change the group name. You go to the link I gave you, or type identity-box-group in the box once you've launched into Special:AllMessages. Then, go to change them (the user right in the right Column is based on what default right used by wikia) and you obviously saw red link because you've never customized them before. Click on the red link (e.g. click identity-box-group-staff if you want to replace the name "staff" into your preferred name) and you click "edit". Destroy the default name for user right and type your preferred name for that individual user right.
I believe TheDeathStar3 is asking about making custom user groups/group rights, not custom user group tags. Unfortunately, Wikia does not allow wikis to customize user groups. If you have a really good reason why you need the defaults changed, you can send a request via Special:Contact/general. However, even with a good reason, they are not likely to allow it.
I believe Sannse has misunderstood what you are asking for; as was the case with Applemasterexpert. The labels you see at the top of userpages are just that; labels. They have no effect on actual privileges afforded to users. The instructions that Sannse and Applemasterexpert gave you are for changing the labels. Most of the time, this is what users are asking about. Combined with the poor English in your first posts on this topic, I fully understand why they may have been confused. It wasn't until #9 that I got an idea of what you were actually asking about.
Another advice: Please include this forum link in your message when sending to Special:Contact/general so that the staff will understand on what you're intention of your question is about (thanks of the clarification from Andrew).