In Into the Woods, there is an interesting split between the first and second acts. In my opinion, the first act is clearly a fairy tale. It follows the stories of Cinderella, Jack and the Beanstalk, LRRH, and Rapunzel to their traditional ends with traditional, happily ever after stories. Even the invented story, that of the baker and his wife, ends happily with a baby. The woods, also, retain the fairy tale motifs of a mysterious place of magic and change. However, the second act departs grievously from the first act. The second act, perhaps because of the loss of the narrator, becomes somewhat chaotic and un-fairy tale-like. Good characters die, which never happens in fairy tales, and the characters are also forced to re-think how they actually liked their happily ever after endings, usually to a negative conclusion. This half of the play ends somewhat unhappily, with everyone having learned a valuable lesson but with sadness and disillusionment all around. So, the first half is definitely a fairy tale, the second half definitely not.
Fables, although it retains all the fairy tale characters, retains them in name only. These characters are far too deep, especially Snow White and the Big Bad (Bigby) Wolf, to be fairy tale characters. The plot, too, is not at all reminiscent of fair tale stories. I would have to conclude very decisively that this, although playing off the fairy tale motifs, is not a fairy tale.