Neapolitan pizza is one thing, Roman pizza is another, and much more akin to New York-style pizza. One type is made in big rectangular pans and sold by weight, it's similar to Focaccia. There is also a round style that is thinner and crispier, which is the best IMO. Fold and eat.
https://en.m.wikipedia.org/wiki/Roman_pizza
While I do enjoy a good old-fashioned Neapolitan pizza, I much prefer the Roman style, specifically the round thin style and by extension the NY style.
Neopolitan pizza has a whole association that certifies restaurants and Lombardi's isn't on that list. It's no surprise that the pizza will a different style.
It's also pretty funny he mentions that in Italy they fold their pizza, when that is practically the defining characteristic of New York style pizza.