Both are untrue.
Vespucci made two trips to the New World as a ship's navigator, the first in 1499. Then in 1503 and 1504 he published two letters he had written to Lorenzo de Medici. In the letters he put forward the idea that what Columbus had discovered was not in fact a new route to Asia, but rather a new continent. Vespucci also published the first letter under the title Novus Mundus, or New World, thereby coining that phrase. The letters were a media hit (but whether because of his innovative navigational theories or his description of the sex lives of American Indians is unknown), and Vespucci became a celebrity.
In 1507, the cartographer Martin Waldseemueller published a map that designated the new world as America and the name was coined. Why Waldseemueller chose to use Vespucci's first name instead of his last is unknown, but it was likely because Amerigo was easily Latinized and the term America shared the same first and last letters as Asia and Africa.
So, America is named after the man who first recognized that it was a new
continent and not just a part of Asia.
Rather fitting actually.