Generally "in the day" is used to describe things that happen during the day. On the day describes events that happen that day.
Examples: On the day of the interview, I wore a blue tie. The event was "the interview".
On the evening of the murder, I was watching TV. The event was "the murder"
I prepare my lessons in the day and give classes in the evening. In the day here describes an action and the time of day.
I eat breakfast in the morning and dinner at night. Again, it describes the action and the time of day the action happens.
I hope this helps!