The first appearance and origin of a Baby Pac-Man is in the third intermission of Ms. Pac-Man. Following the courtship in the second intermission, Pac-Man and Ms. Pac-Man greet the arrival of Baby Pac-Man, who is delivered by the stork. However, this Baby would later develop into Jr. Pac-Man as opposed to Baby, though some home ports feature a "Junior 2" intermission where the stork delivers a second child, presumably Baby.
While Junior went on to get his own game in Jr. Pac-Man, Baby got his own as well, in the form of Baby Pac-Man. This would be the only appearance of the character in a video game until the Pac-Man World series.
Pac-Man World series
Ms. Pac-Man: Maze Madness
Pac-Baby appears in Ms. Pac-Man: Maze Madness in multiplayer mode as a playable character. Once again, she is depicted as female.
Pac 'n RollBaby Pac-Man appears in this game under the name Pac-Baby. Pac-Baby is Ms. Pac-Man's younger sister and the daughter of Pac-Master. She is saved by Pac-Man after he defeats Golvis for the first time.
In other media
Pac-Baby appears in the Pac-Man animated series, voiced by Russi Taylor. He is depicted as male in the cartoon. He is likely to be the character later known as Pac-Man Junior.The likely solution to the male/female discrepancies is that Pac-Baby is a common nickname for infants either in both Pac-Man's and Ms Pac-Man's families, or possibly in general Pac-People culture. The later games featuring Pac-Baby as a girl could take place after Pac-Man Junior's stint as Pac-Baby, and the nickname could have been handed down from one to the other.