1) A standoff. As used today "Mexican" is pointless & the phrase makes no sense. Do you ever wonder why it isn’t called a Canadian or Bulgarian standoff? Also note nearly all the other definitions in this lexicon that start with “Mexican” are unflattering to the culture.
2) Early to mid 20th century S Texas term for a rout. Colorful language used by the profoundly racist (white) Texans of that era. The joke being that in a fight, the Mexican always runs away and later explains that his lack of injury was because "We just stood & stared at each other for a while and then went home."
1)Too silly to use in a sentence.
2)In a scene from "Monty Python and the Holy Grail," the killer rabbit quickly turns the skirmish into a Mexican standoff as the terrified knights obey King Arthur's order to "Run Awaaay!"