There are Jewish-style delis, German-style delis, and Italian-style delis.

What I see being discussed here, for the most part, is an upgrade to a
convenience store sandwich shop.

This is a real deli:


http://www.katzdeli.com/presentation.html

Ratner's Dairy Deli, sadly, is closed.

Carnegie Deli isn't bad:

http://www.carnegiedeli.com/menu.php


Those are delis.