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.
I am certain that "chewy with a shiny exterior" bagels that I get at a bakery are made with yeast. They aren't some sort of secret or rare thing any place I've been, from New York to Arizona.
The mass-produced-in-a-factory-with-lots-of-preservatives bagels (e.g. Lenders) are very different, but also obviously made with yeast, and they don't really resemble fluffy white bread IMO.
When I go to my local grocery store in the northeastern US (the chain is in fact owned by a Jewish family), both kinds are available, the one in the bakery area and the other on the shelves with rolls.
Talking about these as though the dividing line was Jewishness or the use of yeast strikes me as both weird and incorrect.
(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.)