Nakige

This type of a story is designed to have an emotional impact on the player and make itself memorable, crying is usual but mostly optional.

It's explained in wikipedia article about KEY like this:
The developers at Tactics created a simple formula for a game: a comedic first half with a heart-warming romantic middle followed by a tragic separation and finally an emotional reunion formed what is known as a "crying game". The main purpose of such a game is to make the player feel for the characters and make them cry due to emotional scenarios which serves to leave a bigger impact on the player after the game is over.