After the gods leave, Brahma remembers Ratri, who first emanated from him, and tells her that Sati will be reborn to Mena; she is to enter the womb and make the child black, so that Shiva's later teasing sends Parvati to penance and their son can kill Taraka. Ratri is to gain Parvati's qualities and be known as Ekanamsha. She enters Mena's mouth as she grows sleepy at night and stays until the birth. Later Parvati asks Brahma for a golden colour; her dark skin becomes Ratri again, three-eyed, bell in hand, in yellow and red. Brahma gives her the lion born of Parvati's anger and sends her to the Vindhya; the text then calls her Kaushiki.
Brahma remembers Nisha, his own body born formerly, and speaks to her privately. She is to make the child dark, a Kali; after leaving Parvati's body she is to kill the demons led by Chanda and Munda and be known as Chamunda. She enters when Mena's eyes close slightly; the text calls her Vibhavari. Parvati is born at the brahma muhurta. Later a blue-lotus-coloured, three-eyed lady with a bell comes out of Parvati; Brahma gives her the lion and sends her to the Vindhya to kill Shumbha and Nishumbha, called commanders of Taraka's army. She is named Kaushiki.
What is different here:
Names Nisha and Vibhavari, not Ratri or Ekanamsha; foretells Chamunda, Chanda and Munda, Shumbha and Nishumbha; birth at the hour before dawn; no nightfall scene.
When the gods praise the Goddess on the Himalaya, an auspicious goddess springs from the sheath of Parvati's body and is named Kaushiki; after she has come out, Parvati becomes dark and is known as Kalika.
What is different here:
The direction is reversed and Night plays no part.
Sources: The Matsya Puranam, Part II (chapters 129-292), translated by a Taluqdar of Oudh (Sacred Books of the Hindus 17.2, 1917); The Skanda Purana, tr. G. V. Tagare (Motilal Banarsidass), Maheshvara Khanda, Kaumarika Khanda, on wisdomlib.org; The Markandeya Purana, tr. F. E. Pargiter (Bibliotheca Indica, 1904)