What's the difference with a kid killing in self-defense compared to an adult

[Evangeliste]

If a kid plans to kill someone... Even if they don't fully understand it, they sure as hell know it's to cause harm. That's murder.
Self-defence doesn't automatically rule out a guilty conviction when juxtaposed with a death as part of the case. That's where you have things like manslaughter terminology come into play. Usually self-defence comes with manslaughter but self-defence can be used in cases where death doesn't occur like grievous bodily harm and such.

Most bullies are actually opportunistic, they don't plan how they will bully someone, they just go with what's at the time and prey on them when they happen to see them.
They also count on things like fear and embarrassment even tolerance of their victims so they don't get caught.

If the victim happens to be kill their bully at the time of provocation it will still be manslaughter, if they planned to be spotted by the road to push that bully into the road, that's murder.
Whether the court of law can discover the difference, that's down to the victim's mentality... You can see a frightened child to one whose premeditated an attack.

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As long as they haven't returned something they knowingly stole from me, they are (or should be) free game in my books, in- or outside my house.

And that's why you said 'should', that's why they made a law on it. To stop that simple ridiculousness. That and the person you killed, will stop contributing to things you both pay for... Like taxes. Well done, you killed a fellow tax payer.

[Evil Midnight Bomber]

As long as they haven't returned something they knowingly stole from me, they are (or should be) free game in my books, in- or outside my house.

Well, "Your books" don't really amount to much as far as the law is concerned.

[Zafire]

Maybe because it's generally not the case that a child will pose a deadly threat to another child?

It falls under the "children are born pure and good!"-trope.

I've always had a problem with that logic. Kids are generally selfish, greedy, and liars. They dislike sharing. They will do anything to get out of trouble. They'll cry and be obnoxious to get what they want. This isn't always true, but it is most of the time. It makes me wonder if it's true "human nature", we generally get those mindsets hammered out of us as we age. Most of us, of course.

[Deleted]

Somewhat sure that the word bully should be sufficient to explain as to why murder isn't accepted...

[Spiffums]

Usually the zero tolerance policy punishes both bully and victim for "fighting". I have wondered why more bully's don't get a bat to the head to be honest. 9 times out of 10 the bully is older/bigger/stronger than the victim so disparage of force is there. Street thug in school is still a street thug and you can't "open a dialogue and discuss your differences" in the playground anymore than you can in the street at gun/knife point.