Who Discovered What?

There's a story now in The Economist that starts off:

THE brave seamen whose great voyages of exploration opened up the world are iconic figures in European history. Columbus found the New World in 1492; Dias discovered the Cape of Good Hope in 1488; and Magellan set off to circumnavigate the world in 1519. < ... >

It seems more likely that the world and all its continents were discovered by a Chinese admiral named Zheng He, whose fleets roamed the oceans between 1405 and 1435.

The authenticity of the map is being questioned and I'm not following the story that closely, I'm posting here because why does the world continue with this idea that America was "discovered." Sure, someone can have credit for being first for sailing around the world or crossing the ocean or sharing his germs with unsuspecting indigenous populations.

But there were already people here. How can you discover a place that already has people?