To SravanKra: Thank you for your very quick response to my inquiry. The only client I use for my Contacts (Address Book) is iCloud.
But yes, you are surely correct about the 'Custom Labels' being stored 'On the Cloud'. The proof is that the instant I create a Custom Label on one of my Apple devices (MacBook Pro, iPhone or iPad Air), it is instantly synchronized not only with all of my other Apple devices but my iCloud website version of the Address Book, as well. But the problem remains that none of those devices nor the iCloud website offer the capability to SORT the existing unsorted list or even 'DELETE' an item already on the list. Apparently the list is generated spontaneously based on the information currently in my Address Book. But unlike Photos, where DELETING a photo will delete it from all devices, DELETING a Custom Label in the Address Book is a 'Zombie' experience — it's still there the next time you return to the Custom Label List, as long as that Custom Label is assigned to an entry. If, after deleting a custom label, I change the Custom Label to another category, the next time I return to the Custom Label List, it's finally gone. Kind of awkward and arcane to be sure.
All we can hope is that someday the Apple 'Contacts' Designers direct some of their energies in the direction of Custom List management. This problematic situation has been around since the days when 'Contacts' was called 'Address Book'. To use Apple 'Contacts' is to know this problem, so perhaps, those Apple Designers don't really use Apple 'Contacts' themselves? And/or are never involved in the User feedback process?
Although your above suggestion didn't lead to any apparent resolution for me, if you have any other suggestions, I'd welcome them!