Did Columbus want to reach Asia by sailing west or reach Asia by sailing east or discover America or prove that the earth is round?

Columbus, and all educated Europeans of his time, already knew that the Earth was round. That required no voyage for proof.

The trade routes to Asia via the east were controlled by the Turks (via land) and the Portuguese (for the sea route, so Columbus proposed sailing west to reach Asia.

His problem was that his estimate of the size of the Earth was incorrect, so when he reached land in the Caribbean, and later on the American continent, he knew how far he had sailed and thought he had reached Asia.
Columbus, and all educated Europeans of his time, already knew that the Earth was round. That required no voyage for proof.

The trade routes to Asia via the east were controlled by the Turks (via land) and the Portuguese (for the sea route, so Columbus proposed sailing west to reach Asia.

His problem was that his estimate of the size of the Earth was incorrect, so when he reached land in the Caribbean, and later on the American continent, he knew how far he had sailed and thought he had reached Asia.