Everyone agrees on what the Italian word espresso means. The argument is about which meaning made it the name of a coffee.
The word is the past participle of the Italian esprimere, "to express." Etymonline traces it to Latin exprimere, "to press out, squeeze out," built from ex-, "out," and a verb for pressing. English "express" comes from the same Latin.
That gives at least three possible reasons for the name, and Merriam-Webster lists them all. The coffee might be "pressed out," forced through the grounds under pressure. It might be "express," meaning fast. Or it might be coffee made on the spot, expressly for the customer who ordered it.
Etymonline goes with pressure, saying the name refers to the steam pressure used to make it. Merriam-Webster reports that current etymological opinion favors the third reading. Wikipedia notes that the early machines of 1906 took about 45 seconds and made each cup "expressly for you." Speed and personal service fit that history well. More than one sense may have helped the name stick.
The machines came first. Wikipedia says Angelo Moriondo patented a steam-powered coffee machine in Turin in 1884, but it brewed in bulk. Luigi Bezzera filed a patent for single-portion brewing in 1901. Desiderio Pavoni bought Bezzera's patents in 1903 and began selling machines in 1905. The two showed their machines at the Milan fair in 1906.
The drink most people picture today came later. Achille Gaggia registered a piston patent in 1938. In 1947 and 1948 he introduced lever-operated piston machines that produced much higher pressure. They created the foamy layer of crema, and Gaggia sold the result as "crema caffè."
English borrowed the word after the war. Etymonline dates it to 1945, and Merriam-Webster quotes both "caffe espresso" and plain "espresso" in 1958 novels.
The misspelling "expresso" showed up almost at once. Merriam-Webster cites a 1955 New York Times example. It is not simply an error, the dictionary argues, since English already had "express" and the coffee's name comes from the same root. Merriam-Webster enters "expresso" as a variant, calling it "a quirky, jittery" one.
Whichever meaning won, the Latin under both spellings is the same. Something is being pushed out, whether it is coffee, a train or a thought.