Do you think I think the words don't matter in describing those two situations you gave, of course not!
Then why did you treat them that way?
Because your example is completely different. In the original comments case he used murder to describe executions, any reader would still know it is an execution, the word 'murder' just carries emotional connotations in this case. In your example I guess the words you would be referring to is 'I got $10000', so if someone said this to someone they would not know how they got it. If it was option 1 from your post, then the audience would have no idea that it was from bashing an old lady over the head. Or if option two, given the money. That answer is not analogous, it only looks the similar in the most superficial way.
It is interesting that you had to make that stuff up to answer me
It's perfectly reasonable to take your rhetorical construction and to apply it to another scenario so you can see the flaws in what you said.
As I have answered previously, it is not the same. A construct, like a program, does not have to be valid on all inputs. The example you gave, wasn't alike. The construct was not the same either, you replaced it with a generalization for all words, it was specific to that case. I am repeating this but, in the original comment 'murder' carried an emotional connotation, in your example the audience would not have had a clue what really happened.
is totally irrelevant, using the word 'murder' does not change the information someone reading that comment gets from it
Of course it matters. Words mean things. The reason we have different expressions to convey the concept of murder and the concept of the execution of a death sentence carried out against someone who chose to commit murder is: those are not the same things! Labeling them as if they are, and tainting your communication with the connotation of a word chosen when you know it's an inaccurate, agenda-loaded word choice meant to bias understanding of what's said, is not just some breezy situation to dismiss as if it's some linguistic quirk or just the act of someone who's got a childlike vocabulary and doesn't know better.
One chooses the word "murder" to describe an act because one thinks the act is actually murder, and wants to persuade others to perceive it the same way. Don't play dumb like you can't tell the difference.
If you wanted to use neutral language, then you would use 'unlawful deliberate killing' for what the criminal did and 'lawful deliberate killing' for what the state did. But generally people don't talk like that. Most people can cope with communication that has emotional connotations with it.