I think, what we are talking about, are in fact three different things at least. Things, that may be related, but that are still different, and not one and the same. Firstly, the appearance of this phenomenon of altered perception itself, or at all. Secondly, the ability to perform necessary physical acts, and finally, the confidence in one's abilities, you had described, which may lead to some kind of deeper relaxation. I would think, that the ability to control that "level two" emotions, I talked about earlier, also fall into that category.
What you say about an "intense physical art or training" may help in several ways, but it's principal value lies in providing one with a profound arsenal of techniques or movements, from which to draw in those cases of emergency. I think, that belongs to the second and third of the points mentioned above. However, while I think that it may help a lot in such situations, beside being useful in numerous other ways, I'm not even sure if it is really necessary, regarding the things we are discussing now. Perhaps it even might have been something along those lines, that Yagyū Tajima no kami alluded to, when he meant "you need no technical training" (in another version of the story, Yagyū sensei only declared, that the technical instruction would be a very easy and quick task, in consideration of the fact, that the single-most important lesson of the art already was learnt and mastered beforehand).
But surely, I don't know, how my number 6. incident would have ended, hadn't I had years of regularly ukemi-practice behind me. On the other hand, the second incident I wrote about, but without details, included two decisions of performing physical acts, that clearly lay beyond my technical skills, after just a few days of novice-training at skiing. Firstly I had to jump several meters forward with the fastened skis, and then I had to take a 90 degree turn to the left as fast as possible, after landing, without falling down, of course. Both moves worked out, despite of the fact that I hadn't had done such things ever before in my life. And also never again, I would like to add. Also, in incident number four, there wasn't even any chance to meaningful apply any of the physical techniques I train/ed.
However, if any such type of physical training, you're talking about, would be a prerequisite for the phenomenon itself to appear, at least my first two incidents couldn't have happened, as I was only training for a few months then, and that with not more than just one training-session of one hour per week. The phenomenons clearly must have been evoked by something else.
But what kind of thing might this be? I believe, like it is claimed in the legend I quoted, the key is either to have no mortal fear, or at least to have the ability to control fear of death to a high degree. That, at least, prevents the possibility to panic. Panic, I believe, even in a milder form, would hinder the body and mind to change into that state in question, which otherwise might appear naturally. Therefore, I guess, that the most important way to develop the ability to ALLOW that phenomenon to appear, is to already overcome agony by some means or other, before the dangerous situation appears at all. That may be accomplished by different forms of psychical training, by meditation, or even by religious practices. In my case, nothing of that applied. But for reasons, that are beyond this discussion, I wasn't particularly "attached to life", from early age on. For me, that may be an explanation for the first few appearances of the phenomenon in question.
Later, and especially around incident number 4, I already had read and contemplated, what was available to me at that time, like Nitobe's "Bushidō, the soul of Japan", Suzuki's "Zen and Japanese Culture", as well as his "Der Westliche und der Östliche Weg" (I'm not sure if that book appeared in English, too) and other texts. I think, that the reason, such writings appealed to me despite of my relatively young age, was that attitude of defiance of death (but not daredevilry!), that (kind of incidentally) was already present within me.
Now, that I've discussed point two (the means that partly are/were necessary) and point one (how and why the phenomenon might appear at all) of my itemization, point three is still missing. It's the other form of emotion control, that one, that you addressed with the description "confidence in one's abilities".
Until now, you might have asked me: "If you believe, that that altered perception in times of need could only occur, as you didn't really feel mortal fear, how is it possible then, that you, yourself, talked about the fear you felt during some incidents?"
Now, I think, that the type of fear I addressed in point one, is different from the type of fear I talk about now, in point three. It happens on that other, more superficial level, I earlier described, when I talked about some anger I felt during the car accident. The fear I mean now, is a fear happening on that same level. One other example, I can give, is the following: I was, and still am, able to get (sometimes even very) upset about all kinds of bits and bobs. But I'm also able to stay perfectly calm, when it's about more important things. I think, that physical training can help to become able to control emotions appearing at that level. I don't think, that it necessarily has to be budō training, but I also don't think, that it doesn't matter at all, which kind of training it is. Every training might address it's own goals. Budō training, naturally, will especially increase the "confidence in one's ability", regarding physical altercations with other persons, or even animals. That is my explanation, why I had felt that kind of fear during the incidents that happened at the begin of my training, but not during later confrontations. I even suspect, that that might have been the reason, why my body didn't "feel the need" to change into that "special mode" at occasions, where, to me, it would have been "rational" to do so. But, perhaps, the level of physical harassment I had perceived unconsciously, may simply haven't been "high enough".
Please be aware of the fact, that everything I wrote now and earlier in this thread - and it was not easy for me, to discuss this in English, I hope I was able to find expressions and wordings, that made clear what I mean, most of the times - are only my opinions, based on the personal experiences I've made, and my thoughts about them. I have no idea, if any of that opinions is scientifically tenable.