How do historians determine the beginning and the end of the various ages in Middle Earth? What defines, say, the border between the Second Age to the Third (for example)?

The Ages (of which there are Four containing all the Races) are decided along events as well as dates. Before the First Age was a time of great conflict of the Valar. The First Age begain when the strife ended. The Fouth was after the end of the War of the Ring.

What decided the Second and Third are a few key events:

The Third Age begin with the first defeat of Sauron by Isildur.

It ends when Bilibo, Frodo, etc leave from the Grey Havens.

The Second Age started with the banishment of the Dark Lord Morgoth and ended with Akallabêth, the downfall of Númenor.