The story
First use appears c. 1876 in the US. Three-way gun standoffs, popularized in spaghetti westerns such as The Good, The Bad and The Ugly (1966), have come to be called Mexican standoffs, though this usage appears to date to the 1990s, notably in reference to Reservoir Dogs (1992); earlier usage refers to this as a three-way standoff or triangular standoff.
Etymology from Wiktionary, CC BY-SA 4.0.