If you don't want to use $login_name, then don't register it.
You should be using the $SESSION[] (or $HTTP_SESSION_VARS[]) array anyway - for safety and performance reasons. But then, if you don't want to use $SESSION['login_name'], then don't.
Either way, there's no reason I can see to use two sessions. Especially if you've got some need to switch from one to the other.
If you want certain pages to look different depending on whether the login is public or administrator, then just use a session variable and store some appropriate value in it. $_SESSION['login_type']=='public' or 'admin' or whatever.