You are right, but "western" as its used colloqially doesn't really mean anything.

The term "western" has multiple origins, but the two dominant ones are: (1) European (especially british) terminology, from 17th-19th century history and political science (2) Cold War concepts where the world is divided into 1st, 2nd and 3rd world countries depending on political alignment.

West European narratives from the late British, French & Dutch Empires goes roughly like this.

(1) Classical greece is the "craddle" of (western) civilization. (2) Greeks spread "the light" worldwide with alexander, What he called hellenization they call westernisation (3) Rome eventually took over. (4) The pure values of the west empire was degraded by the decedant east. The (wrestern) roman empire fell and the dark age began. (5) The light of the west was rekindled in the Renaissance (6) The torch passed to west european empires, who dutifully picked up the white man's burden, spreading civilization and light to the world again. (7) britain passed the torch to America, which is Athens' true succesor.

This is the basic narrative running through most 17th-19th century anglo-american history and philosophy. It's comically self serving. So bogus it's hard to know where to start.

First, Greece was never "Europe," in any sense that includes britain, germany, etc.. Europe remained tribal and "uncivilized" (non-urban) for 2,000 years after the Classical Greek civilization spread far and wide. Greece (and after that, Rome) was a place the western periphery of the Mediterranean and middle eastern cultural complex. This region including lots of long lived, large & complex civilizations: Egyptian, Bablylonian, Hittite & Persian empires. Advanced Phoenician city states. Greece's urban culutre fits squarely into this wider culuture. Nothing western about it.

This would not have been controversial to plato, alexander or julio cesar. They looked to egypt, persia and the "east" as the predecessor and contemporary civilizations. They credit Egypt constantly. They didn't know or care about europe as we think of it, and certainly didn't consider themselves part of it.

Second.... this kept happening in every age. The European dark ages actually ended when Europeans regained cultural contact with the "east," via the arab expansion into western Europe and the crusades.

Basically, the concept of "western civilization" is a farce. There is no such thing. It's a false narrative about the origins of european civilization, giving it an indigenous origin. It was just to embarrassing to admit that Europe was tribal until 600-700 years ago while the east had been "civilized" for thousands of years.

Later on (now) it gets equated with democracy, rule of law and such, the political ideals of western europe and the US. None of these originate in the west. None of these are even traditional in the west. American democracy is no more descendant from Athens' than a hundred other ancient city states' political systems. They just happened to know about athenian democracy, through an accident of history.

These days, "western values" has come to mean feminism, liberalism, pluralism, gay rights... Anyone over 30 can tell you that these are not traditional western values. These are modern, cosmoplitan values and they thrive in the places which are not traditional.

TLDR, Western values roughly mean descendant from greece, according to silly 18th century british intellectuals. Mexico is (culturally) more descendant from Greece than the US or Brittain. Basically, it's BS. Western doesn't mean anything.