Murder is a synonym for kill but you can differentiate between them to make a point that one particular instance of such a caused-death is worse. Is more reprehensible.
The semantics of the word are as fluid as the opinions of those who you are trying to explain the situation to, using such distinctions.
If you think the death was wrong, it is a murder. If you think the death was right, it was a murder, killing, assassination, or any such word. Language is obviously not as black and white as the example I gave, but the point stands.
I agree with your definition but think it’s too narrow, and thus missing the point of the original argument. I don’t agree with lo_zamoysk‘s original point. I think lions CAN murder. I think when they commit cannibalism it’s only when they murder other lions. All other deaths lions cause, lion or other animal, are killings (maybe murder maybe not). But when Lion A kills and eats Lion B, Lion A would have much preferred to get food another way. It’s a lot more likely Lion A is motivated by something other than hunger, like so many of Lion A’s - or even any Lion’s - kills are.
Motivations are required for murder. The word “murder” ascribes motivation to a killing.