Is it such a big drama? The Wellington Phoenix are the team from
New Zealand so in some sense they are the New Zealand (based) team
in this competition. Just like the Victory are the Melbourne team.
I would object to the use of "The New Zealand Phoenix" as that is
an error and after a certain period of time it becomes a deliberate
(or at best ignorant) misuse of the name.
Of course as the local fans we are protective of the
Wellington part. That puts the responsibility on us (club and
supporters) to make the Wellington part a distinctive and
important feature of the Phoenix "brand". If the club can
establish a good name (reputation) for itself then all that
nonsense about the actual name will take care of itself. Let the
people making the mistake know about it and give them a chance to
correct themselves. Isn't that what you would want?