Vadrik, Astral Archmage

If a triggered ability triggers whenever day becomes night or night becomes day, it only triggers if it was already either day/night and becomes the other. It does not trigger when the game becomes day or night for the first time.

Before a player untaps their permanents during the untap step, the game checks to see if the day/night designation should change.

All costs to cast a spell are locked in before mana abilities are activated and those costs are paid. If Vadrik's power changes after costs are locked in, or if Vadrik leaves the battlefield while you are activating mana abilities or paying those costs, the cost reduction effect won't be affected and the total cost to cast the spell won't change.

If it is night, and the active player of the previous turn cast two or more spells during their turn, it becomes day.

If it is day, and the active player of the previous turn cast no spells during their turn, it becomes night.

Day and night are designations that the game itself can have. The game starts as neither. Once the game becomes day (or less commonly, night), the game will be exactly one of them—day or night—going back and forth for the rest of the game.