"For the life of me, I can't figure out why in USA Police officers go immediately to murder &....."

[Saviolo]

17. Regardless what the rules of engagement are

In the military, you follow those rules of engagement, though.

As you said:

In other words, there was no one set of rules of engagement. They varied by place; they varied over time.

The difficulty in the USA right now is that the situation is unclear and nebulous. There is a theoretical set of rules of engagement between police and citizens, but how do we know when they're being broken? To my (admittedly untrained) eye, it would seem that the shooting of Tamir Rice, the officers over-reacted, made no effort to de-escalate, and shot the boy within 2 seconds of arrival. Neither of the officers involved in that incident were indicted, so it would seem that gunning down a 12-year-old without any attempt at de-escalation, or determining whether the gun is real, prior to firing is within those rules of engagement.

There are plenty of other examples, we even have a few here in Toronto. Sammy Yatim, Andrew Loku, Jermaine Carby. And for each of these shootings, yes, there are dozens of police interactions that go smoothly. The trouble is not that -all- cops are bad or trigger happy, the trouble is that -any- cop -could- be, and there' no way for someone to know in advance if this officer is nervous, or afraid of POC, or like the officer that shot Tamir Rice, that the force:

[the officer] was unable to follow "basic functions as instructed". [The police chief] specifically cited a "dangerous loss of composure" that occurred in a weapons training exercise, during which Loehmann's weapons handling was "dismal" and he became visibly "distracted and weepy" as a result of relationship problems. The memo concluded, "Individually, these events would not be considered major situations, but when taken together they show a pattern of a lack of maturity, indiscretion and not following instructions

If we're going to encourage the militarization of the police in North America, then it will become necessary to also have a military level of discipline and obedience to strict orders. If the citizens don't know the rules by which LEOs are operating, then there will be no way for people (and at a higher rate POC) to know what to do. If Philando Castile can comply with the officers involved and still be shot while following their orders, there's a missing link between what the rules of engagement are, and what we're being told they are in the streets.