This is a great but also huge question, spanning centuries and continents, so I'll mostly limit my answer to early modern Latin America: hopefully also a starting point for more discussion. So I'm basically turning the question around to look at how some European terms for the Americas were created in opposition to Europe

To start I'll add a few points on your question:

First, as you say the idea of the West in opposition two other parts of the world is highly political, and so tied in part to European colonization and racial hierarchies. These are no "neutral" geographical distinctions (if those exist) - after all usually former European settler colonies with major European influence like the US, Canada and Australia are seen as part of the West; while other types of colonies eg in Latin America and Africa are not.

Second the ideas and concepts I describe are still highly influential: still in the 1960s pre-Hispanic art was usually described as "primitive ", similar to African art (opening debates on how exclusive ideas of art are, but here I'm venturing too far from my expertise). And still today encyclopedias of philosophy tend to have virtually nothing on pre-Hispanic American thought. So concepts like the West are also tied to hierarchies of knowledge, and long-standing ideas that non-Western regions are somehow incapable or truly producing art, history and knowledge, when nothing could be further from the truth.

Right let's rewind to the 16th century.

Early Spanish terms for the Americas

(Adapted from an earlier answer.)

I'll first look more generally at the term America and similar terms in 16th century Europe, and from there move to the Las Indias concept in early modern Spain.

America/ Nuevo mundo

Edmundo O'Gorman has traced the concept of Columbus having found a new part of the world. He famously argued that it is not possible to "discover" something ex nihilo that was already there. For this retroactive process of describing America as a continent only "waiting to be discovered" he coined the term "invention of America". O'Gorman traced the de development of such ideas from Columbus through various writers influenced by him, including Las Casas and Columbus' son, to become highly influential on later perceptions of America.

I've already mentioned the official term "Las Indias" in Spain, but of course there was a multitude of other terms floating around in Europe and Spain in the 16th century, often with no clearly fixed meaning - other examples include "Nuevo Mundo" (New World), following Amerigo Vespucci, and "The Fourth Part of the World". Specifically for "America", we know the term was introduced by Martin Waldseemüller in his famous map of 1507 (map link - it has "America" written on today's South America).

The map built on Vespucci's name and a letter from 1503, now known as "Mundus novus", in which he argued that the land mass spanning to the south of Cuba was not the Asian coast but an unknown part of the world - crucially he called it "new" instead of "unknown", contributing to the "invention of America" mentioned above. Although Waldseemüller would later change his view of the newness of America, the damage had been done so to speak. Sebastian Sobecki sums up some of the following developments and terms in use:

The term gained momentum only during the second half of the sixteenth century, following Gerard Mercator’s (1512–1594) projections of 1538. The spatial and narrative invention of the American continents remained an unfinished project well into the seventeenth century, despite the circumnavigations of Ferdinand Magellan (c.1480–1521) in 1519–1522 and Francis Drake (1540–96) in 1577–1580. Spain did not adopt the name “America” until 1758, and English writers, usually drawing on inferior, homemade cartography, continued to struggle with the name and the newness of the continents.

John Dee (1527–1609), self-appointed cartographer and imperial ideologue to Elizabeth I, insisted as late as 1580 on the term “Atlantis” for the Americas, and in 1595, Walter Raleigh (1554–1618), having trusted Dee’s maps and advice during earlier attempts to colonize Virginia, went searching for El Dorado, a bizarre conflation of distorted Spanish reports of Muisca traditions and faint echoes of India’s famed gold.

We can note here that not only in Spain but also in England (and presumably other European countries) there were various terms for what we know usually call America, and that their definitions were often far from clear and could incorporate mythical elements. Speaking on the "New Found Land" (nicely ambiguous term), Sobecki notes that in England prior to English colonial endeavours, it was often seen as a

no-place, not an exotic world but a forlorn outpost in the cold North Atlantic. These works do not testify to any noticeable interest in the New World or, for that matter, new worlds; they notice the latest additions to speculative geography in passing and with indifference.

While specific to England, we can see similarly how in the early 16th century Spain a lack of knowledge of the Americas would have led to confusion regarding terminology. Only towards the mid-16th century do we find detailed maps depicting the Americas, such as Sebastian Münster's Novus orbis. According to Walter Mignolo "It was toward 1555 that the world began to look to our hypothetical European observer very much as it does today for many people on this planet."

This knowledge included Spain, where Charles V had given the quite accurate Agnese map of ca. 1546 to his son Philipp II, who by the mid 16th century ruled over the Spanish overseas territories (here's a link to the map by Battista Agnese, which looks certainly more accurate than Waldseemüller's earlier one). But while the term "America" can be found in various Spanish literary and cartographic works before the mid-18th century, it did not hold an equal status to the "official" Las Indias.

Interestingly, the term America and especially the form „americanos“ (Americans“) had become important by then in Spanish America - even before the time when the term "Las Indias" was officially abolished. Leading up to and esp. during the wars of independence of the early 19th century, creole elites would refer to themselves as „americanos“ as a way to distinguish themselves from the Spaniards (often called „gachupines“ pejoritively). So at least by this time, I would say that the term americanos resonated more in Spanish America than in the Spanish metropolis. This would only increase with the French coinage of „Latin America“, as a tactic to highlight French influence in the region.

Not to make this even more confusing, but in my readings of native colonial Mexican authors (16th/17th centuries), the term „Nuevo Mundo“ also comes up sometimes. This would have been taken from European authors of course, but was used by some of these Mexican authors to refer to „this New World“, in order to show that the pre-colonial history of „their“ continent (sometimes meaning just Mexico) was in many ways equal to the „Old World“. As I mentioned, the term New World goes back to Vespucci's notions of having "discovered" a heretofore unknown part of the world - e.g. Waldseemüller similarly talks of a Terra Nova in this context.

I'm very late. /u/drylaw gave a magnificent answer regarding early América, particularly Latin América. I'd like to add my two cents, taking a look at something different: the contemporary geopolitical meaning of the "West". I'll use both historiographical and IR and Poli Sci theory resources, so bear with me on the theory part. This answer mainly focuses on the second part of your question, and seeks to answer if, at least within geopolitics and IR, is indeed still in place. Spoiler: it's very close to no longer being in place.

The West, in geopolitical terms, exists nowadays only in certain theoretical frameworks, particularly neo-institutionalism, as well as certain attempts to reinvigorate structuralism. Both approaches have something important in common: they tend to rely on empiricist, quantitative methodologies, and therefore consider, in the pursuit of objectivity and statistical analysis, that there is still a need to divide the world into categories.

However, that kind of approach has been challenged in recent decades by many theoretical approaches, mainly the ones we call unaligned or subaltern schools of thought. What are they unaligned from? Eurocentrism.

International relations theory, as most social sciences, changed drastically during the second half of the XX century, due to several factors, including but not limited to, postmodernism and globalisation.

Postmodernism, the post-structuralist and post-Marxist conceptualisations of theorists such as Roland Barthes, Jacques Derrida, Michel Foucault, Ernesto Laclau and Chantal Mouffe (who is usually credited as a co-author in his husband Laclau's works, but has many fascinating works of her own, I highly recommend them), caused many IR theorists and historians to change their approach, because postmodern thinking proposed the revising of traditional scholarship, particularly regarding the State, democracy, and foreign affairs.

Globalisation, on its part, caused a change in the traditional IR paradigm, because it led us to ask ourselves if concepts such as Westphalian Sovereignty, balance of power and even peace in its traditional definition, were enough to analyze the changing geopolitical arena. The nineties saw the emergence of new challenges for internationalists and political scientists, due to the increasing importance of phenomena such as unprecedented migratory waves: instead of the early waves that saw Europeans migrate towards colonies, migration was now reversed: the consequences of colonialism and war (among other factors) caused many people to escape the periphery towards Europe and North América. I won't delve deeper into this however, because it would take me past the twenty year limit, since we all know that these migrations continue today.

During the cold war, "West" had two meanings in IR and geopolitics: the traditional, historical definition, and the capitalism-communism dichotomy. After the Berlin Wall fell, after the perestroika and the glasnost, coupled with the advent of the internet and mass media, the transnational world that authors such as Nicos Poulantzas and Joseph Schumpeter had envisioned was no longer as clear-cut, because they hadn't foreseen first, the new policies that neoliberalism applied in a globalising planet, and second, the fact that the dissolution of the USSR didn't have the expected outcome.

Henry Kissinger, one of the most well known figures in the US' IR history, spent many years consolidating the United States' power, in order to achieve one very specific goal: when the cold war inevitably ended and, with it, the bipolar world order, the US would be able to govern a unipolar world, as the hegemonic superpower. This has not been the case, and, despite attempts by conservative thinkers to argue different, several things challenged this idea. Deng Xiaoping's reforms positioned China as one of the biggest economies in the planet. The conformation of several international institutions and economic alliances outside of Europe and the US such as BRICS (Brazil, Russia, India, China and South Africa), Mercosur (Argentina, Brazil, Uruguay, Paraguay and Uruguay), the Common Market for Eastern and Southern Africa, to name a few, led the world increasingly towards transnationalism as the core of foreign policy.

To add to these factors, theorist and philosopher Immanuel Wallerstein stated in several works during the 80s, 90s and onwards, that we no longer live in a discerningly polarized world, but rather in a modern world-system, in which abstract concepts (in terms of political scientist Giovanni Sartori) such as State and nation are no longer the only actors in the geopolitical stage, because the nineties saw the consolidation of multinational corporations that possess as much economic power as many countries.

Another important factor that should be taken into account postcolonialism and decolonial studies, schools of thought that seek to validate unaligned, native ways to do social sciences. Most of these ideas promote the existence and furthering of new, anti-eurocentric philosophy, cultural anthropology, sociology, Poli Sci and IR, that "belong" to nations and regions outside the European and American spheres, thus challenging the traditional concept of the "West". There are many proposers of such subaltern theories, such as Eduardo O'Gorman (whom drylaw mentioned), Enrique Dussel, Silvia Rivera Cusicanqui, Boaventura de Sousa Santos, Achille Mbembe and Gayatri Chakravorty Spivak. They tend to be supporters of regionalist sociopolitical and economic models such as pan-Africanism and the Latin American Patria Grande or Nuestramérica (Great Country or Our América). Those who have read my answers may have noticed that I distinctly write América instead of America, because to us Latinxs, the continent is called América, instead of the English concept of the Americas. That cultural difference is a part of the more subtle yet important factors that, coupled with those I stated before, have led in the past thirty years to a reexamination of the traditional distinction between the West and the rest of the world.

References:

De Sousa Santos, B. (2015) Epistemologies of the South: Justice Against Epistemicide. Routledge.

Mbembe, A. (2001) On the Postcolony. Stanford: University of California Press.

Sartori, G. (1998) Homo Videns. Buenos Aires, Taurus.

Spivak, G. C. (1988) Can the subaltern speak? Basingstoke: Macmillan

Wallerstein, I. (1991) Geopolitics and Geoculture: Essays on the Changing World-System. Cambridge: Cambridge University Press

Wallerstein, I.; Lemert, C.; Aguirre Rojas, C. A. (2013) Uncertain Worlds: World-Systems Analysis in Changing Times. Boulder, CO: Paradigm Publishers.