If you take a look at start.php in mod/groups there is a chuck of code that sets up the side menu.  You would want to put a check around this to make sure that the user is an admin.

     add_submenu_item(elgg_echo('groups:new'), $CONFIG->wwwroot."pg/groups/new/", '1groupslinks');

The name could be changed in the language file but from the sounds of what you want to do, you are going to want to change the name based on some type of value that is assigned to the group.  This value would identify if it was a group or a tribe.

If the Admin makes the groups public then anyone will be able to join these groups.

Hope this helps.

Kathy

To rename groups to something else you can use the following plugin:

communities

Hope that helps.