Does at midnight mean midnight exactly?

Probably the "Midnight Ride" of Paul Revere in 1775, the American Revolution, to warn re the British Army (or else Tam O'Shanter by poet Robbie Burns when he escaped from the witches!)....

It means, "in the middle of the night."

PM stands for past meridiane

Staying awake late into the night and losing sleep. You'll be burning the midnight oil the night before a final exam. Many U.S. taxpayers are burning the midnight oil on April 14th.

It shows that Bella was Edward's light in his life even at night, where they usually saw each other.