I have read that when a man kills his wife or son, or when a ruler kills someone while disciplining them, then there is no need for Diyah.

Which type of discipline is meant here? For me it sounds like an injustice because I think disciplining somebody to the extend that he/she dies is not correct.

Also, wasn't it the rule that legal execution should only be done by the state and not by individuals, now how come that a man has got the right to kill his family due to discipline?

Kindly clarify this issue for me.

The discipline meant here is the normal way of disciplining. So we already know that part of the father's duties is assuring the child is disciplined and this sometimes might include corporal punishment (of course taking into consideration principles like, it's not about you hurting the child but showing her/him that what they did is wrong, the minimum force possible is to be used). If however while disciplining the child he/she sometimes dies then there is no diyah for many reasons including the fact that there is no intention to kill as well as it's accidental: the father was after all doing what is the shariah permitted him to do.