I've wondered about this question, and
have come to a tentative answer. I'll have to check with a real
historian who knows the city to verify my guess, but until then,
I'm convinced by the internal evidence.
Under the old political system, Newark had many small, numbered
wards (as opposed to the few large, geographically named ones it
has now). Ward 12 was Down Neck and the Island. Dutch Neck was part
of Ward 11. There were many more people living in Down Neck than
on The Island, so basically Ward 12 was Down Neck. Even after a
large part of Dutch Neck was transferred into Ward 12, Down Neck
politicians dominated the Ward, and so in many minds, "Down
Neck" and Ward 12 were the same. This was probably the beginning.
After the reorganization, I believe (This is one of the points
I really need to speak to an historian about) that former Ward 12
politicians, especially Down Neckers, dominated East Ward politics,
and so again all of the territory they controlled came to be considered
part of Down Neck.