I love how people think a western is a cowboy, an indian, and a frigging cactus, out in the middle of the desert. No. No. And No.
That was just the "American" culture during the Old West (I really despise calling it the "Wild West", sounds too corny).
What makes a western a "western", is that it falls under the following criteria:
1. It is set in the Western US (including Alaska (Those are called Northerns), Canada, Mexico, The Southeastern US, and Australia. And some countries in Europe/Asia during the 1800s-1900s had many outlaw gangs (especially in Italy Spain. Turkey, and the Middle East). Meaning, you do not need a "cowboy", as there are no video games where you play as a cowboy (gunfighters, lawmen, and other citizens).
2. Characters that are in a "grey" zone in terms of morality.
3. A sense of lawlessness, guns, money, drugs, crime, murder, etc. (This is what makes the "Mexico" trilogy, NCFOM, and CoJ: The Cartel (and yes, I do like this game, and anyone that disagrees with me needs to stop basing their gaming decisions on reviews), a western).
4. Law. Law enforcement.
If these criteria are met, a western can be set in any time period. However is is preferabbly 1836-1920. Anytime from 1920-Present Day would make it a Neo-Western (which is still a western).
But yeah, I prefer Mexico as a settng for westerns, as it was (and still is) one of the most violent areas in the world.