Midnight is obviously 12:00am, but people don't know which day it belongs to. Is midnight the start of the day? Or the end of the day? Tuesday 12:00am could be 2 different times (spaced 24 hours apart) depending on your interpretation. If you say 12:01am, there is no ambiguity.

A lot of people in high school probably ignored their math teacher when he explained open/closed intervals.