The Don did not like making open threats to people and generally did so only as a last resort or when he felt the situation was personal (the bandleader story). Evidently he trained Tom and Sonny and Michael to follow in his footsteps in that regard.

Tom points out to Woltz that he hasn't threatened him and thinks to himself that the Don taught him that way. Even after Sonny loses his temper and beats Carlo and threatens to kill him it is explained that if Sonny really meant to do murder he never would have said so. Michael remains calm in the face of Green's insults and implied threats and makes no counter threats. Though he could coerce Jules' or Lucy's help as front people in Vegas he doesn't do so.

The Don is said to constantly be reminding Sonny of the foolishness of expressing anger. When talking to Michael about the utility of someone like Luca, he speaks dismissively of Luca's tendencies towards rage.

Why did the Don feel so strongly about this? Was it the experience of seeing his (presumably) rash father and brother murdered? Was it just good business sense? Was it because once you make a threat you may have to follow through or lose face?

I think Vito just realized that threats are usually empty, and that actions speak louder than words. And although wiretaps and listening devices were almost unheard of in the '40s, it's always better to not incriminate yourself if you can help it.

Lilo, the novel states that, indeed, it was his father's murder (that followed upon the heels of his being so verbose with the local mafia chief) that prompted Vito's avoidance of quarrels, threats, etc. It also states that Vito taught Tom to never threaten, to reason only.

In general, isn't losing one's temper in public perceived as a sign of weakness?

Think being calm in the face of confrontation is in Vito's nature.

There is story in the novel about Vito working somewhere with Irish workers and them heaping ALL kinds of verbal abuse at him daily, and Vito pretending to not understand what they were saying and just calmly finishing his job.

I think his "management style" of never making a threat developed as he ascended in the underworld and dealt with others who were hot headed by nature or felt that making threats helped them achieve whatever they were trying to get done.

He saw the error of using such tactics,and being a businessman on top of everything, he always sought a way to RESOLVE issues or disputes in as constructive a manner as possible.

He might have arrived at the conclusion somewhere in his criminal life that it was his father's hot heat that lead to his demise, but he surely wasn't old enough at the time to understand that.

Tom doesn't threaten Woltz openly, but in fact he does: He's talking about labor trouble and a film star using heroin.
Woltz hits the nail on the point: “I understood perfectly,” he said. “That’s the Mafia style, isn’t is?"

Their style is to let their opponent not lose their face.

That wasn't a threat. That was a friendly offer of assistance to correct a possible issue.

But you're dead on about giving the other guy a way to climb down from a hard position without losing face. Win-win situation for everyone.

He didn't even want to see Luca privately at the wedding. But was sure to call him his "most valued friend" when they did meet briefly.

He knew how to handle the bulldog.

As the novel states, Luca was the only one who could make Vito nervous.

Another reason Puzo stated was that Vito wanted his enemies to underestimate him, the better for them to be incautious. Vito's not answering threats, insults, bullying, etc., by others would have the effect of making them let their guard down.

And speaking of making threats:
Michael obviously hadn't learned from his father when he was holed up in Sicily. He tells Sr. Vitale who he is, what he's doing there, and then says that some people would pay a lot for that info but then his daughter would lose a father instead of gaining a husband. Not too swift...

Chalk it up to his inexperience TB.

Seriously though, you are right. And his boasting that info to Vitelli only supports my post over in the GF Trilogy pertaining to how Michael sat there and bragged to Kay about the power that his father and his family had. Again, as far as I am concerned that line to Kay, (paraphrasing) "It's my family Kay, it's not me" is a bunch of bullshit. I firmly believe that while wanting to be his own man, deep down there was something in Michael that aroused him in knowing how powerful his father was and how if he chose to he could one day wield that same power!