Murder is defined as "the unlawful killing of another human being". The standard of "unlawful" is relatively defined. It's murder when a suicide bomber, mass shooter and serial killer deliberately targets innocent people, but it's not unlawful when you kill during a war, assassinate a tyrant or put someone to death as punishment.
Every definition I have read include qualifiers like "unlawful" and "unjustifiable".
Yes, whether an act is murder/justifiable tends to be objective. The victims and those close to them would likely see the act as murder despite public or court belief and the perpetrator and others would likely believe their actions were justified, again, despite public or court belief. Either way, this does not change the definition of the word "murder".