You are mistaken. The law does not say "all killing is murder."
Num 35:22-25 However, if he pushes him suddenly without enmity, or throws anything at him without lying in wait, or uses a stone, by which a man could die, throwing it at him without seeing him, so that he dies, while he was not his enemy or seeking his harm, then the congregation shall judge between the manslayer and the avenger of blood according to these judgments. So the congregation shall deliver the manslayer from the hand of the avenger of blood, and the congregation shall return him to the city of refuge where he had fled, and he shall remain there until the death of the high priest who was anointed with the holy oil.
So the Law makes a clear distinction between murder and manslaughter. The murderer is put to death but the manslayer is not.
A further distinction is made by God when He sends Israel out to make war on their enemies.
Deu 7:2 and when the LORD your God delivers them over to you, you shall conquer them and utterly destroy them.
Deu 20:16-17 But of the cities of these peoples which the LORD your God gives you as an inheritance, you shall let nothing that breathes remain alive, but you shall utterly destroy them: the Hittite and the Amorite and the Canaanite and the Perizzite and the Hivite and the Jebusite, just as the LORD your God has commanded you,
And the rest of Jesus' statement is: "
But I say to you that whoever is angry with his brother without a cause shall be in danger of the judgment. And whoever says to his brother, ‘Raca!’ shall be in danger of the council. But whoever says, ‘You fool!’ shall be in danger of hell fire."
It is the main point of the book of Esther.
All the Jews defended themselves from the attempt to exterminate them.
In the book of Judges, when Israel was attacked by their enemies, their Judges called for repentance and then they went out and slaughtered their enemies.
David, the "man after God's own heart" slaughtered the enemies of Israel.
Judith slew Holofernes to protect the Jews.
Jael slew Sisera the commander of the enemies of Israel.
None of these were considered guilty of murder.
So there is very definitely a difference between killing and murder.