The author of the book seems to be confounding two different categories. The excerpt is about how there was no state apparatus to investigate homicide as a matter of course. But that doesn't mean there was no laws against homicide.
Eventually it would be extended to homicide outside one's family (Lex Cornelia de sicariis et veneficis), but as the GP says the article confuses things a bit between the state not being involved at all, and a lack of investigation.
I think murder was just like anything else in that regard, meaning that the state would not prosecute any crimes on its own without someone going to court in the first place. But that was also the case for the jewellery example, or anything else: if nobody complained about it, the state did not have a police force looking out for these things and the crime would go unpunished.