Baxter Stockman's story in the 2003 games mostly matches up with the 2003 cartoon, save for a few details. Baxter creates the Turtlebot in the 2003 video game, unleashing it to smear the Turtles' name and defeat them (in the cartoon, the Turtlebot wouldn't show up until much later). Baxter also functions as a boss, in his mecha suit, in Donatello's path.
Stockman shows up mostly in cutscenes in the following two games, and appears in the Game Boy Advance version of Teenage Mutant Ninja Turtles 2: Battle Nexus as the next to last boss. He also appears as a boss in Teenage Mutant Ninja Turtles: Arcade Attack, using his Back to the Sewer design.