Saraswati is born of the Goddess and given to Narayana
How the Shakta Puranas tell Saraswati's birth: a white daughter from the tongue of the Goddess (or Radha's lips, or Krishna's mouth), with a veena and a book, whom Krishna sends to Vaikuntha as Narayana's wife, and whom he worships himself on the fifth day of Magha.
Narada asks Narayana about Prakriti; Narayana names her five forms, and Saraswati is the third.
At the start of creation Krishna divides into a male right half and a female left half, the Goddess Prakriti.
From the tongue of the Goddess comes a white daughter, dressed in white, with a veena and a book, presiding over every shastra.
The Goddess divides into Lakshmi and Radha; Krishna divides into a two-armed form and the four-armed Narayana.
Krishna tells Saraswati to follow the four-armed one as his wife, and Lakshmi the same; Radha stays with him.
Saraswati wishes to stay near Krishna; he tells her Radha is the deity of his life and sends her to Vaikuntha, where she will have its Lord as husband.
Narayana takes Lakshmi and Saraswati to Vaikuntha.
On the fifth day of bright Magha, the day learning begins, Krishna himself worships her; after him Brahma, Vishnu, Shiva, the sages, kings and all people.
Narayana tells Narada that Mula Prakriti, Krishna's left half, puts out five forms for creation, Durga, Lakshmi, Saraswati, Savitri and Radha; Saraswati is the third, the goddess of speech, learning and music who lives as intelligence in every heart and without whom the Brahmins would be dumb, and Brahma was the first to worship her when she was born. Chapter 2 gives the birth: after a hundred manvantaras a white daughter comes out of the tongue of the Goddess, in white, with a lute and a book, mistress of every shastra; the Goddess divides into Lakshmi and Radha, Krishna into a two-armed form and the four-armed Narayana, and Krishna tells Saraswati to follow the four-armed one as his wife; Narayana takes Lakshmi and Saraswati to Vaikuntha. Chapter 4 adds that she sprang from the end of Radha's lips and wished to stay with Krishna, who told her it would do her no good, since Radha was the deity of his life, and sent her to Vaikuntha to have its Lord as her husband; and that Krishna himself worshipped her on the fifth day of the bright half of Magha, the day learning begins, after which Brahma, Vishnu, Shiva, Ananta, Dharma, the sages, kings and all people did the same, with white offerings, a jar, Ganesha first, and the dhyana of a white, smiling goddess with lute and book.
After the dissolution Krishna is alone. From his right side come the three gunas and then Narayana, who praises him; Shiva from his left side with five heads; Brahma from the lotus at his navel with four; Dharma from his chest. Then Saraswati appears from Krishna's mouth with a veena and a book, radiant like crores of moons, lotus-eyed, in pure white, smiling, the mother of the Shrutis, the shastras and the learned. Lakshmi comes from his mind, Mula Prakriti from his intellect, Savitri from the tip of his tongue (crystal-white, with a rosary, the mother of the Vedas), Radha from his left side. He gives Lakshmi to Narayana, Savitri to Brahma, Durga to Shiva and Rati to Kama; Saraswati is not in the list, though Brahma is later called her husband. Chapter 8 names the five Prakritis and makes Saraswati the deity of intellect, speech, learning, poetry, memory, imagination, authorship, music and rhythm; chapter 9 fixes her worship on Magha Shukla Panchami, the day of Vidyarambha, with white offerings and the dhyana of a white goddess with book and veena, and says Krishna first taught her kavacha in Goloka.
What is different here:
From Krishna's mouth, not the Goddess's tongue; Savitri, not Saraswati, from the tongue; no scene of her wishing to stay with Krishna; she is given to nobody in the handover list; the gods arise from Krishna's sides and navel rather than from a divided Prakriti.
Bhagavata Purana 3.12.28-34 (the daughter named Vak; not told)
Pan-IndiaVaishnava
The older birth, recorded here only as the contrast: Brahma has a daughter named Vak, born from his body; his mind turns to her; his sons led by Marichi tell him that no Brahma before him has done such a thing and that it does not suit one whom people follow; ashamed, he gives up that body, which spreads in every direction as fog in darkness; then the four Vedas come from his four mouths. Left out of the site under the family rule; the Vedas from the mouths are named in the god page's roots.
What is different here:
Brahma's daughter, not the Goddess's; shame and an abandoned body; no veena, no Narayana, no Vaikuntha.