It would be the 12th day waning crescent moon.
From the links shared in the question body it is clear that moon came to reside on Shiva's head to cool down the effect from Kālakūṭa poison that he had devoured. This happened on Ekadashi, and next day moon came out
From Chp 10 of Brahma Kanda, Padma Purana
Then on the twelfth day in the morning, when the sun had risen, the great Lakṣmī, graced with all (auspicious) characteristics, sprang up. All the religious deities saw that great mother of all creatures having her abode in the heart of Viṣṇu, and were delighted. Then, the cool-rayed one (i.e. the Moon), the brother of Lakṣmī, sprang up with nectar. (So also) Tulasī, Viṣṇu’s wife, purifying the world, came up.