I agree with your definition of murder except for the term, "innocent". It is not the role of a private individual to make such judgments. This is the role of the civil magistrate, not an individual without authority to do so. There is no problem with killing another person in self-defense or in protection of other people when they are imminently threatened with death or injury. Jesus Christ said in Luke 22:36: "But now, whoever has a money belt is to take it along, likewise also a bag, and whoever has no sword is to sell his coat and buy one." The sole purpose of the sword is protecting self and property, not acting as a self-appointed executioner.
Besides, the murder of Tiller did not prevent one abortion. His death will not stope even one of some 1.5 million babies from being killed by abortionists this year. If anything, the act of unauthorized vengeance will be counterproductive, for even if the Left is unsuccessful in using his death to push pro-abort and anti-Christian legislation, it will persuade many people to reject the pro-life argument. If abortion is ever to end in this country, we need to win the hearts and minds of the opponents and the uncommitted.
You don't know how many murders his death will prevent. Given the small number of doctors willing to murder babies in their third tri-mester, I would assume it would prevent some murders.
If the third tri-mester fetus cannot be called an innocent life, who can?
I did ask the moral question, do we not have an obligation to prevent those murders?