What led to the switch from "Settling America" to "New World" to "the Americas?"

This is really building off of a conversation I had in a different thread in this sub. We all know that the first widely used term for the landmasses of North, Central, and South America was the New World. Then, in the 18th Century, the term Two Americas was coined. Even more recently, the term Old World is used to refer to the continents of Asia, Europe, Africa, and Australia. We also now use the term the Americas to refer to Central, North, and South America. I'm curious as to what led to the slow shift from "New World" to "the Americas" and what led to the widespread adoption of the Old World term as well.