It might be my stupidity to think that many laymen terms that most people use to describe some physics phenomena usually have a scientifically accepted term or name? The process of star formation, does it have a specific name generally used in the scientific word? Or is Star Formation used widely by scientists and the rest of us?
An ADS search for "star formation" turns up about 142,000 articles with "star formation" in the title or abstract. The first article is a 43 page review paper of Star Formation in Galaxies in the Hubble Sequence, written by Robert Kennicutt, Jr, one of the leaders of the field. He never defines anything else to mean star formation and one of the "key words" used for tagging the article is "star formation."
I think it's safe to say, "star formation" is the technical term that everyone uses. Note, though, that as an extension to Jim's comment, star formation is generally used in the context of a region of space that is currently forming stars, not for a single star that is being formed.