Surely the ancient world was more common, with bandits, etc, but do you have any justification that murder was a 'personal thing'? I imagine most communities had was to prevent / punish this.
I think in general, for antique societies, they don't have detectives - if a crime is committed, the first problem is, there's no state agency interested in investigating the matter.
The claim that murder was a 'private matter' is from Emma Southon, but it makes a lot of sense to me. Pre-modern history is rife with blood feuds that are basically the 'private' way of preventing murder. The state typically wouldn't have the resources to get involved, even if they did have the interest.
If you imagine the chain of somebody_is_murdered to somebody_is_punished there are questions like: 'who reports the crime, and to whom?' and 'how do you establish the crime actually happened, in the way it was reported?' that are just really hard to answer unless you have a big state organization dedicated to managing stuff like that.
I suspect the differentiation they are drawing is that in "modern" society there are very few accepted reasons for killing someone, and all cases should be examined by an arms-length person.
This is very much not true across societies, or historically.
In the Ancient world, I don't expect murder of slaves or subsistence farmers would have any involvement outside of their families/owners/patrons.
It's fortunate that no action is required from the murder victim, I guess.