There are clear rules about using lethal force in the United States which are the same for police as anyone else. The major difference is police are required to engage in situations that necessitate it rather than required to retreat from them.
Unfortunately the SWATer knew exactly what to say to provide the requirements for lethal force. Jeopardy, (I will kill you), capability (I have a gun), and Imminence (I will shoot them when they come out.)
The police had reasonable cause to believe the call was legitimate. Any reasonable person would have believed the same, given the information they were given and training they have gone through. Just because the guy wasn't pointing a gun at them didn't mean he couldn't do so before they can react. If you have time you can see demos and videos of situations exactly like this one that end with police and innocent bystanders being killed.
The blame lies with the SWATer himself and the systems that failed to recognize that the call was bogus. We can't blame police for failing to recognize that this situation wasn't like the ten others that they or their coworkers responded to where the man was armed and the kids were dead.