I can't speak to the other continents, but have written about how the idea of "America" became popular over here. The term came up in the early 16th century in connection with Amerigo Vespucci, and knowledge of the "new" continent became more widespread from the mid-16th century onwards - so much later than terms for other continents like Europa and Asia. We can also keep in mind that such ideas of e.g. America as a specific continent are from a European perspective - would be fascinating to learn about such ideas of continents from other cultures/traditions.

I can also recommend this answer by u/terminus-trantor for more background on Vespucci and America.

This is the relevant part from my earlier answer (for the links to maps see the original thread):

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.

While interesting conceptually, your question deals more with the term America and its use in Europe and/or Spain.

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.

[Edit: This part from the same answer seems also relevant here, as it mentions the idea of a "New World" going back to European perceptions of Europe in the 13th century, meaning a perception of Asia as a different part of the world:]

I should add that some of these terms draw on larger European traditions. I mentioned "Las Indias" having been used for Asia prior to the 15th century, especially following Marco Polo's voyages. Similarly, the idea of a mundus novus or new world for Asia can be traced to the 13th century. Variations of this phrase were often used in the following centuries for describing encounters with Asia. So we can note that some terms such as Las Indias and New World draw on European traditions related to Asia; and that the idea of a "discovery" of America created a further tradition of incorporating the Americas into European ways of thinking. This would influence changing European perceptions of what "world" and "cosmos" could mean in the 16th and 17th centuries.