In the months before the death, Livingstone had twice breached a court-ordered protection order preventing him from contacting Ms Webb.
Police granted Livingstone diversion for the first of those offences, against national policy.
"The prosecutors were acting on the mistaken belief that a prosecutor could authorise diversion for minor family violence matters to a particular level," Mr Coster said.
"It is accepted that the prosecutor was wrong."
Following that mistake, a national audit was conducted and uncovered six occasions when diversion was granted for protection order breaches.
Mr Coster had overseen changes as a result of the audit.
They included the creation of an adult sexual assault squad in Dunedin, a more objective approach to investigations and a willingness to progress investigations involving reluctant victims.
"I am confident that the Southern police district now has the structure and processes in place to ensure family violence and sexual assault victims receive the highest level of service," he told Chief Coroner Judge Deborah Marshall.
Earlier in the day, Detective Senior Sergeant Kallum Croudis said multiple failings contributed to a situation where Livingstone was able to kill his children.
Anne Stevens, counsel for Ms Webb, asked Mr Croudis if Dunedin police's handling of family violence involving Livingstone reflected a "succession of failures".
"That's true," he replied.
In May 2013, Livingstone trapped Ms Webb in their bedroom and raped her over a period of five hours, even as their distraught daughter banged on the door.
Police were made aware of that information when they attended an incident at the residence on May 27, 2013.
Mrs Stevens asked if Livingstone could have been imprisoned for rape if police had handled the investigation better, Mr Croudis replied: "It's possible - it's probable, even".
The protection order was first issued on May 31, 2013, and finalised on August 13.
On August 7, Livingstone breached the order for the first time.
While reporting the matter to police, Ms Webb presented a number of empty bullet cartridges which Livingstone had given to Bradley and Ellen during an earlier supervised visit.
Police did not take the matter further - a decision which was flawed, Mr Croudis said.
Livingstone had collected the cartridges when he sighted a rifle with his one-time flatmate Philip Mans.
Ultimately, it was one of Mr Mans' firearms that Livingstone stole to kill his children.
It was unclear which child was shot first, but Ellen died from a single shotgun wound, while three discharges were apparent in Bradley's body, Mr Croudis told the court.
Ms Webb became distressed as details of the children's deaths were read out.
The hearing continues today.
