The gods and their enemies had agreed to churn the ocean together for the drink of deathlessness, with the mountain as their rod and the serpent as their rope. What came up first was not nectar. A poison rose that burned in every direction, and the creatures of the three worlds began to die of it. The gods ran to the mountain where Shiva sat with his consort and sang his greatness. He turned to Bhavani and said that a lord's task is to shelter the living, even at his own cost, and asked whether she would let him. She gave her consent. He took the whole poison into the hollow of his hand and swallowed it. It stopped in his throat and stained it blue, and from that day the singers call him Neelakantha. A few drops had spilled while he drank; the scorpion, the cobra and the poison plants took those. The Bhagavata gives the verses (8.7.18–46) and places the scene on Kailash.
The Himalaya · Pankaja–Madhumati confluence, UttarakhandShaiva
The temple thirty-two kilometres above Rishikesh, where two hill streams meet, says this is the ground on which Shiva sat to drink. Its tower is carved with the churning; pilgrims fill it at Shivaratri and through the month of Shravan, when the Kanwar carriers bring Ganga water to cool the throat that held the poison.
What is different here:
Localises the act to a Garhwal hillside and ties it to the Shravan water-offering.
Among the reasons for the night vigil of Shivaratri is that on this night Shiva drank the halahala and held it in his throat, which swelled and turned blue; devotees keep watch through the night as the gods once did to keep him awake and the poison from sinking.
What is different here:
Dated to Shivaratri; the all-night vigil is explained by it.
In the telling most people know, Parvati catches Shiva by the throat as he swallows so that the poison can go no further, which is why it stays there; and the cool crescent moon is placed on his head to soothe the burning. Neither detail is in the Bhagavata verses fetched.
What is different here:
Adds Parvati's grip and the moon as coolant; oral layer, unverified against any Purana.