Actually, both you and the article are correct. From Wikipedia: "When the electoral ballots were opened and counted on February 11, 1801, it turned out that the certificate of election from Georgia was defective: while it was clear that the electors had cast their votes for Jefferson and Burr, the certificate did not take the constitutionally mandated form of a 'List of all the Persons voted for, and of the Number of Votes for each'. Vice President Jefferson, who was counting the votes in his role as President of the Senate, immediately counted the votes from Georgia as votes for Jefferson and Burr, and no objections were raised."

HOWEVER, after he counted all those voted, he and Burr were tied (at that point, first place was President and second place was Vice President). So, there was a contingent election to decide the President after that, with Jefferson tying Burr for the first thirty-five times until Hamilton convinced some Federalists to switch their votes to Jefferson.

TL;DR: The article is correct that the President of the Senate chooses which votes to cast. Jefferson chose to open and count Georgia's certificate, but that left a tie between him and Burr, leading to a contingent election.