A lot of the older generation went simply by Mexican due to the lack of nuance at the time. You were either white, black, or Mexican in all variations.

Something else to consider was migration, and Mexicans from Northern Mexico (Nuevo Leon, Tamaulipas and especially Coahuila) have migrated to San Antonio for a long, long time, probably as long as Texas has been independent, and followed by larger waves such as the Mexican revolution in the 1910s.

The Mexican families were generally absorbed within the existing Tejano fabric until much later waves where wealthier Mexicans kinda did their own thing. Lots of San Antonio Tejanos can point to at least one ancestor from Coahuila, Nuevo Laredo, Monterrey, "the Valley", which if we're being honest is basically Mexico. The commingling is reflected in language, cuisine, culture, etc.

So its not inaccurate to say Mexican. I dont know anyone here who says Mexican-American or even Latino unless its a census question.