When a movie is based on an earlier work (book, play, or remade movie) the WGA makes decisions about how much writing credit to adapt from the original work. In this case, the history is as follows:
Roi Cooper Magrue wrote the play Seven Chances.
David Belasco rewrote it for Broadway and produced it on Broadway.
Magrue wrote a movie adaptation for Buster Keaton, basing it on his original script. Keaton was not satisfied with the movie script and had Clyde Bruckman, Jean Havez, and Joe Mitchell rewrite, to bring it closer to the Broadway play. This ma have been a commercial decision, instead of a creative one, owing to the Belasco's well-known success as producer versus Magrue's relative obscurity as a writer.
Because of the movie rewrite, Magrue was not given the screenwriting credit, but was given the adaptation credit, and a secondary playwrite edit. The rewrites brought the movie closer to the Broadway version, so Keaton gave Belasco primary playwright credit, and Magrue got a secondary playwright credit (after Belasco) and the adaptation credit. At this point the tables are somewhat tirned, sice for Broadway, it was Belasco who adapted from Magrue.
Steve Cohen rewrote Buster Keaton's movie for its remake, "The Bachelor."
Because of the series of rewites going from play scripy to Broadway to movie script to rewrite to remake, it appears the WGA decided that David Belasco's Broadway play was sufficientl far removed from the remade movie that he was not entitled to credit for it. Nevertheless, since the earliest version of this work is Magrue's original play, Magrue is still given a playwrite credit in The Bachelor, for the PLAY "Seven Chances." Oddly, this mans that an intermediate work that influenced the first movie is left out of the credits, while an earlier work is included.
So, yes Virginia, David Belasco did write some version of the original, but the movie is deemed so far from his work that he is disassociated from it.