New York style pizza is frickin horrible. those thin floppy slices can't support any reasonable amount of toppings. I actually prefer Domino's over most New York style pizza, and Domino's is pretty bottom of the barrel. The worst offender is the New York style cheese pizza which is basically just an oversized soggy triangular cracker which appears to have been dipped in oil. Blech.
But outside of New England "New York Bagel" is polite code for Jewish bagel, ie the type using unleavened bread. It means actual bagel, not bagel-shaped bread.
(I say jewish rather than kosher, because many of the best bagel places don't qualify as kosher. At Vancouver's best bagel place (siegels) you can get cheese on your Montreal smoked meat bagel.)
All bagels are leavened (real bagels with yeast, whether commercial or sourdough, but I'd be surprised if even 1% of bad bagels are made with non-yeast leavening). The difference between an airy bagel and a tough bagel is the type of flour (high gluten) and the cooking process (boiled then baked).
A Jewish bagel weighs about three times as much as a non-Jewish bagel, and is about three times as dense. It doesn't have the airy, bread-like consistency that a non-Jewish bagel has. As my Jewish friends like to say (paraphrasing), "If I can throw it at your head and risk knocking you out, then it's a Jewish bagel."
I've never had a bagel that was "airy", but I've also never had one that was made without yeast. They're normally pretty chewy, but definitely not unleavened. You boil and then bake them.
No, they're referring to real bagels (chewy, with a shiny exterior) as "Jewish", as opposed to fluffy-white-bread-roll-with-a-hole that is sometimes marketed as a bagel.