Downtown is a term primarily used in North America by English speakers to refer to a city's core (or center) or central business district

In the U.S. it means whatever part of town is considered the center or core. Maybe like "manzana urbana?" It is not restricted to big cities. Our little city of 40,000 has a "downtown." It doesn't have a precise definition. Usually, the historically oldest part of a city is also the financial center, has the tallest skyscrapers, etc. In Petula Clark's popular song hit of 1964, "Downtown," she uses it to mean the place with the "night life," clubs, bars, movies, theatres, etc. I don't know why we say DOWNtown. My guess--well, it's the same as what Wikipedia says: "The term is thought to have been coined in New York City, where it was in use by the 1830s to refer to the original town at the southern tip of the island of Manhattan."

Downtown / Civic Center = El centro financiero de una ciudad grande

Can't resist adding this one, referring to the Madison, Wisconsin, pop. 240,000 "Downtown is actually up on a hill… Your walk up State Street will lead you eventually to the Capitol Square, site of the Wisconsin State Capitol." This "downtown" is NOT a financial district: "The State Street pedestrian mall links the University of Wisconsin campus with downtown Madison. A walk up State Street will acquaint you with all kinds of ethnic cafes, restaurants, coffeehouses and shops. Marvel at the Overture Center for the Arts.... Museum corner is at the top of State Street with the Children’s Museum, the Wisconsin Historical Museum, and the Wisconsin Veterans Museum."

Examples: "To-morrow he would go into the roaring downtown district and find work."--O. Henry, referring to New York "Close to the low-rise structure are all the exciting attractions of downtown San Antonio, including the Alamo, River Walk, Rivercenter Mall, and Henry B. Gonzales Convention Center."--Guidebook, referring to a city of 1.4 million "The downtown district is actually a few blocks east of White Sands boulevard, off 10th street."--Guidebook, referring to Alamagordo, New Mexico--a city of only 30,000 "The economic heart, the so-called downtown where I worked, was actually crowded against the western edge of the metropolis, along the Santa Cruz riverbed..."--Tucson, Arizona "Indianapolis's initial recreation and culture program involved a set of new facilities in a lengthy (east-west) linear arts district in an area four miles north of the decaying downtown area."