Your definition of murder is out of line with the meaning as used by the rest of society, as defined in the dictionary:
https://www.merriam-webster.com/dictionary/murder
Definition of murder
: the crime of unlawfully killing a person especially with malice aforethought
was convicted of murder
2 a : something very difficult or dangerous
the traffic was murder
carrying the luggage was murder on my back
b : something outrageous or blameworthy
getting away with murder
3 : a flock of crows
There's a reason the proper term for a flock of them is a murder of crows, and it's not because we like having them around. —Jeffrey Kluger
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and, my views are in line, not unsurprisingly, with those of the Catholic Church as written in its Catechism:
http://www.scborromeo.org/ccc/p3s2c2a5.htm#2267
2266 The efforts of the state to curb the spread of behavior harmful to people's rights and to the basic rules of civil society correspond to the requirement of safeguarding the common good. Legitimate public authority has the right and duty to inflict punishment proportionate to the gravity of the offense. Punishment has the primary aim of redressing the disorder introduced by the offense. When it is willingly accepted by the guilty party, it assumes the value of expiation. Punishment then, in addition to defending public order and protecting people's safety, has a medicinal purpose: as far as possible, it must contribute to the correction of the guilty party.67
2267 Assuming that the guilty party's identity and responsibility have been fully determined, the traditional teaching of the Church does not exclude recourse to the death penalty, if this is the only possible way of effectively defending human lives against the unjust aggressor.
If, however, non-lethal means are sufficient to defend and protect people's safety from the aggressor, authority will limit itself to such means, as these are more in keeping with the concrete conditions of the common good and more in conformity to the dignity of the human person.
Today, in fact, as a consequence of the possibilities which the state has for effectively preventing crime, by rendering one who has committed an offense incapable of doing harm - without definitely taking away from him the possibility of redeeming himself - the cases in which the execution of the offender is an absolute necessity "are very rare, if not practically nonexistent."68