A war is usually the total annihilation of the poeples adhering to the identity of the group you have a grudge with.
You're the one who doesn't understand war, then. What you're describing is genocide.
Genocide is one of the form that can take war, but you can culturally subjugate, or use political and/or religious indoctrination. It's the identity that you must erase, not the individual.
Edit : of course terrifying the opposing group and killing its core members is usually the first step. Execute it's priest, sages, politicians,.... or subjugate them.
Look at the Romans, Huns or English strategy.
War is an organized, armed, and often a prolonged conflict that is carried on between states, nations, or other parties typified by extreme aggression, social disruption, and usually high mortality. War should be understood as an actual, intentional and widespread armed conflict between political communities, and therefore is defined as a form of political violence. The set of techniques used by a group to carry out war is known as warfare. An absence of war (and other violence) is usually called peace.
Straight from Dr. Wikipedia.
War is not the extermination or eradication of an identity, group of people or group of identities. War is the physical combat carried out to further political machinations.
The Romans usually left locals in charge of their own government, but just added their 'Governors' above the local politicians. They also usually tried to get the locals to join their auxiliaries, to promote a sense of unity with the Roman people.
The Huns did the same. They came in, asked for tribute and deference in exchange for near-autonomy, and if this was not achieved, they butchered and destroyed as a warning to those who would not choose to join them. They did not execute politicians or local seers or wisemen. They left those men in their positions of power, and provided a liason to keep the community in line with the Huns expectations.
I don't know about the English.
The fact of the matter is, most successful empires did NOT kill enemy leaders. They embraced them, and USED them to keep the locals in line. It is far easier to control someone when the person they look up to is telling them to do as you say, than to try to control them through brute force. People resent brute force deeply.
I'm not sure where you got your ideas of what war is, but they are not exactly correct.