The Joe version would seem to violate the cappuccino standards put forth by the Specialty Coffee Association of America (S.C.A.A.) and its Barista Guild, which advocate a one-centimeter layer, minimum, of milky foam.

Only one centimeter? Sounds dangerously close to a latte. But who would know better than the S.C.A.A.? “It’s kind of ridiculous,” said David Schomer, the founder of Espresso Vivace in Seattle.

Some coffee specialists pointed to “latte art creep” as responsible for the small amount of foam in the modern-day cappuccino, noting that it is easier for baristas to make intricate designs with less froth in a time of Instagram-ready food and drink.

Given the changes in what constitutes a cappuccino, some people may find themselves with an attachment to an incarnation of the drink that was in style when they came of coffee-drinking age. “Back in 1985, the best cappuccino was the one with five-inch mounds of froth sprinkled with cinnamon,” the restaurateur Daniel Meyer wrote by email. “We gave up on foam in 2006.”

Mr. Carmichael of La Colombe recalled the cappuccino at an influential cafe in Seattle, Torrefazione Italia, long before specialty coffee drinks were common. “Cappuccino was coffee with really thick meringue-type foam,” he said. “You could set an olive on it and it wouldn’t sink.”

Kenneth Nye, who founded the East Village cafe Ninth Street Espresso in 2001, grew so sick of customers’ insistence on what they believed to be a “real” cappuccino that he removed all the drink names from his menus. “All it says is ‘espresso with milk,’ ” Mr. Nye said. “We stopped with the names because it’s all silly.”