Similar rules for Screen Actors Guild. Michael J Fox is really Michael Andrew Fox, but there was an actor already registered as Michael Fox. Michael J Fox chose the middle initial of "J" for his SAG registration.
Vanessa Williams story from Wikapedia:
Williams is most often referenced and publicly recognized simply as "Vanessa Williams." There is, however, occasional confusion with similarly named actress Vanessa A. Williams, who is just two months younger. It has been reported that Williams first became aware of Vanessa A. in the 1980s when her New York University registrar told her that another, similarly aged student with the same name and from the same state had applied.[13][14] When Williams appeared as Miss America in a Macy's Thanksgiving Day Parade, Vanessa A. accidentally received her check for the appearance, which she returned.[13]
In the area of acting, the two ran into name conflict when Screen Actors Guild rules prohibited duplicate stage naming. Vanessa A. had registered the name "Vanessa Williams" first,[13] so as a compromise, Williams was occasionally credited as "Vanessa L. Williams" in acting credits. To compound the confusion, both actresses starred in versions of the drama Soul Food (Williams in the film version, and Vanessa A. in its TV series adaptation). The Screen Actors Guild eventually took the issue to arbitration and decided that both actresses could use the professional name "Vanessa Williams"