"Tex-Mex” (never “Mex-Tex") became popular in the 1960s as the name of a cuisine.
(Oxford English Dictionary)
Tex-Mex, a. and n.
Also Texmex. A Texan style of cooking using Mexican ingredients, and characterized by the adaptation of Mexican dishes, frequently with more moderate use of hot flavourings such as chilli; food cooked in this style.
1963 N.Y. Times Mag. 11 Aug. 50/1 Star of the evening was her Texas or Tex-Mex chili. 1966 Great Bend (Kansas) Daily Tribune 19 Oct. 5/4 It’s too bad that it has become known as ‘chilli powder’ because some homemakers may associate it only with the preparation of ‘Tex-Mex’ dishes. 1988 Independent 4 Aug. 20/5 We..eat Texmex, half steaks and salad and ice-cream, and half..cheese and tortillas.
26 December 1952, Galveston (TX) Daily News, pg. 12 ad:
TOMATO PUREE 3 4 1/4-Oz. Cans 10c
10 November 1957, Galveston (TX) News-Tribune, Cook Book, pg. 17:
13 October 1960, Delta Democrat-Times (Greenville, MS), pg. 14?:
Adds Savory Touch
“Tex-Mex cookery” is the name often given to that savory style of cooking, using meat, beans, tomatoes, onions and green peppers, smartened with chili powder, oregano and other spices. Chili con carne, enchiladas and tacos top the list of Southwestern favorites so often teamed with thin corn pancakes—tamales—the daily bread of old Mexico.
5 October 1961, Dallas Morning News, “Tolbert’s Texas” by Frank X. Tolbert, section 4, pg. 1:
MRS. FRANZ STUMPF of San Antonio, a published authority on what she calls “Tex Mex food,” has also taken this department mildly to task about the greaseless (or nearly so) chili recipe I’ve been offering.
29 April 1962, The Daily Texan (Austin, TX), “Our Nearest Neighbors...Mexico” by Debbie Howell, pg. 2, col. 6:
Mexican food, across the border. is very different from the Tex-Mex type served here. It s light, not at all greasy, and delicious. Tacos, enchiladas, chalupas, and the already known Mexican foods are considered merely appetizers or antojitos (little whims).
Google News Archive
24 May 1962, Pittsburgh (PA) Post-Gazette, “Dangers of Eating at the Fair (Seattle World’s Fair” by Douglass Welch, pg. 45, col. 2:
SO WE WERE GOING by this exotic Mexican place, and everyone here says it is genuine Mexican cuisine as contrasted with Tex-Mex which is what you get in California and along the southern border.
11 August 1963, New York (NY) Times, “Never Too Hot for Chili” by Craig Claiborne, pg. 214:
Star of the evening was her Texas or Tex-Mex chili.
3 November 1964, Oneonta (NY) Star, pg. 11:
Some of the best Mexican food in New York is served in Greenwich Village on Charles St. at a simple neighborhood type family restaurant called El Charro. Garcia, behind the bar, or his wife, Maria, will shake you up the craziest Marguerita in town (tequilla with lime and cointreau) and their enchiladas, chilequillos, tostados and tacos are of the famous Tex-Mex variety so revered by Southwesterners.
7 October 1965, Elyria (Ohio) Chronicle-Telegram, pg. 17:
You can do all kinds of delicious things with chili powder, even though it is most famous as the seasoning for “Tex-Mex” dishes—chili con carne, enchiladas, tacos, refried beans, et cetera.
13 January 1966, Dallas Morning News, section E, pg. 11:
You can do all kinds of delicious things with chili powder, even though it is most famous as the seasoning for “Tex-Mex” dishes—chili con carne, enchiladas, tacos, refried beans, etc.
21 July 1966, New Mexican (Santa Fe, NM), pg. 15:
“A Bowl of Red” by Frank X. Tolbert (Doubleday) is about the tonsil-searing gourmet food of the Southwest which few people from less sturdy regions have ever been able to swallow—chili con carne, tamales, tacos, enchiladas, jalapeno cornbread and son-of-a-bitch stew. The Tex-Mex food is a state of mind; you have to be born in the area to understand it.
7 March 1969, New York (NY) Times, pg. 32:
Dining Out Where the Menu Is “Tex-Mex”
By CRAIG CLAIBORNE
And it has Tex-Mex food—that highly seasoned and imaginative Texas version of Mexican cooking—of a very high order.
23 May 1971, Austin (TX) American-Statesman, “How to ‘Talk Texan’” by Wendy Haskell Meyer, magazine, pg. 10, col. 4:
If you really want to join the in-group, pick up the Tex-Mex cooking techniques (cheat a little and buy a tortilla maker) and keep a stock or jalapenos and okra on hand.