He was with them 1997-2000. Then he was put on waivers and the Red Sox claimed him. His career essentially had collapsed. Suddenly, he was “Big Papi”!
http://www.baseball-reference.com/players/o/ortizda01.shtml
He was with the Twins through 2002, not 2000. He wasn’t put on waivers, they had the option to offer him arbitration or allow him to become a free agent. They chose the latter.