Caffe Reggio claims to have introduced the cappuccino to the United States. When Domenico Parisi set up shop in New York’s Greenwich Village in 1927, he whipped up foam-crowned cups for delighted crowds. Word of this Continental pick-me-up spread and soon cappuccinos were cropping up on menus across the country. The espresso machine, which dates back to 1902, is the oldest on display in the city. In addition to this retired workhorse, the cafe contains an original bench from the Palazzo Medici Riccardi in Florence. Besides being the place where the…