death penalty

But you forget that in biblical times there were quite a few crimes with a penalty of being stoned to death, which was the penalty that God set. I really don't think that executing someone by proper authorities as the result of a crime counts as murder.

Right.

The state following its own rules which are known beforehand != murder.

Breaking those rules by killing another = murder, and is punished by the government for that specific crime.

Not all murders are death-penalty worthy, though. Killing mentally retarded people is just wrong. They didn't know that they were breaking any rules, and they wouldn't understand why what they did is wrong. Jail for them, to keep them from the rest of society.