Is it because of James Madisons urging, the election of 1800 was finally settled? or was it someone else?

Election of 1800

Connecticut, Delaware, Georgia, Kentucky, Maryland, Massachusetts, New Hampshire

New Jersey, New York, North Carolina, Pennsylvania, Rhode Island, Tennessee,

South Carolina, Vermont, Virginia

Candidates Thomas Jefferson Aaron Burr John Adams

Political Party Democratic-Republican Democratic-Republican Federalist

Electoral Vote 73 73 65

States 8 8 7

House Run-Off 10 4 -

Adams in the end gave the election to Jefferson, who had tied with Burr.

so it's Thomas Jefferson correct?

No, Adams settled it. See the last sentence.

Yes, you are correct. In the election of 1800, it was ultimately John Adams who settled the tie between Thomas Jefferson and Aaron Burr. After the Electoral College vote resulted in a tie between Jefferson and Burr, the decision was sent to the House of Representatives to break the deadlock. Adams, being the incumbent president and a member of the Federalist Party, chose to give his support to Jefferson, who belonged to the Democratic-Republican Party. This endorsement by Adams led to the House choosing Jefferson as the third President of the United States.