Intent can be thought simply as being similar to wanting to pick up a glass [the intent or meaning behind the action] and the body responding without having to consciously think about it. Training allows us to do actions without having to focus our attention on the steps required. So, what you talk about with yi being in-between the conscious and unconscious minds is perhaps OK; and that with training we have developed a “neural map” or “primed our nerves” (they already know how to accomplish the task) which allows us to perform the actions without devoting conscious thought to them is also probably OK. But these things are not really special; science already knows about these things and they are not exclusive to martial arts (indeed, everybody does these things every day).
What makes it seem more special is that various demonstrations, like those you do, illustrate how we seem to overcome resistance differently when physically trying to oppose force vs. the effective use of what seems to be less/minimal effort (i.e., using intent instead of strength).
So, how do we explain the difference between ineffective physical strength, and when “moving with intent” rather than with strength seems to be able to succeed? This I do not think that science has studied well. I address some related concepts in the following article:
http://slantedflying.com/what-the-unbendable-arm-can-reveal-about-taijiquan/More specific to your demonstrations, I would speculate as follows:
Under stress we tend to initially tense up and contract our postures (to prepare for subsequent actions). We can see this if we watch what happens when someone is startled. When not in contact with a potential opponent this response is probably OK. But once we are in contact with an opponent, and dealing with their force/pressure on us, that instinctive contractive response is counterproductive. We want to train our bodies to respond differently than that instinctive tensing reaction. The tensing leads us to use the wrong muscles and therefore they subsequently require us to “force” things. In TJQ the quality we seek is peng (like a properly inflated ball that receives force without collapsing or pushing back; or like a pine tree limb receiving snow).
So what does physical effort do differently than when moving with intent? When we try to lift up against resistance we tend to FLEX our arm; but when using “intent” instead, we EXTEND the arm. Flexing uses the yin muscles that are better designed for absorbing force rather than projecting force, which is the function of the extensors or yang muscles. When someone uses “muscular strength” to push against resistance from one side to the other, we tend to lead with the body and this again FLEXES our arm rather than EXTENDING the arm as is done when one instead utilizes trained “intent’ to move the arm. Rather than using the biceps, we should be using the triceps and the muscles of the back (and, by extension, all the way through our structure and into our feet which push against the ground, i.e., whole body power instead of isolated arm strength).
Unless trained to maintain the potential to change from absorbing to projecting (using yin muscles then yang muscles) and vice versa, people typically react to failed actions by trying to add even more of the same thing. If pushing does not initially work, then push harder! I would argue that, instead, we should push DIFFERENTLY. It is difficult for most people to change actions when what is required is opposite the initial one. How we initiate an action can be the most important factor, especially when acting from contact and under force/pressure from an opponent. We should train to avoid the drawing in reflex unless we are deliberately using it to absorb and pull the opponent.