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 exo suit, in Donatello's path.
In the 2003 GBA game Stockman appears in cutscenes. Mirroring his plotline from the first 2003 cartoon he creates the Mousers and Foot Tech Ninja. If either one of these stages is cleared, Stockman's dialogue sprites will appear with a missing/scarred eye for the rest of the game (for that Save file). This means he will have a scarred eye during Leonardo's story on replays or if Donatello's story was cleared first. This also means that if Donatello's story is played first before Leonardo's story, Stockman won't yet have a scarred eye in Donatello's story cutscenes.
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. In the DS version of Teenage Mutant Ninja Turtles 3: Mutant Nightmare, Stockman appears during Donatello's path where he helps him break into Shredder's lab. He also appears as a boss in Teenage Mutant Ninja Turtles: Arcade Attack, using his Back to the Sewer design.
Though he is nowhere within the plot of Teenage Mutant Ninja Turtles 3: Mutant Nightmare, he does appear in an unlockable self-voiced movie within the game's Theater section. He appears in his cybernetic wheelchair with one eye scarred and a neck brace, while inside Stocktronics. Using one of his computers while he provides narration, he displays images of his genius invention of the Foot Tech Ninja.