* “Mexican” is an odd designation. Sometimes it has the traditional meaning: a person who either lives/lived in Mexico or is directly descended from the people(s)—note that plural—of Mexico. Most often in present times in the US, it’s a catch-all term for any person of Hispanic descent (even though a great many such people come—especially in recent years—from places like El Salvador, Guatemala, Ecuador, and Chile; also Cuba, but Cubans are geographically localized and have a strong community with a distinct identity).
For what it’s worth, my nine-times great-grandfather traveled from Toledo, Spain, to Mexico City in 1679. His two sons traveled north to Santa Fe and settled there in 1705, and the family pretty much stayed in that area for the next 250 years. So . . . in 1679, Mexico (the entire territory, which included most of present-day New Mexico) was (politically) New Spain, and its inhabitants (whatever their genetic makeup) were citizens of Spain.
They stayed Spaniards until the end of the War of Mexican Independence, in 1821. At this point, my paternal ancestors became Mexicans—for the next thirty years, at which point the Treaty of Guadalupe-Hidalgo (1848) ceded the New Mexico Territory to the US, and we became Americans, all without moving a step. (My maternal ancestors—mostly English, with one odd German branch—didn’t arrive in America until the 1700s, the laggards. . . .)