...I am using the word, as perhaps coined in print, by the Dorrance brothers and Ray Hunt, to mean being physically attuned to the horse's mental and physical stater, aware of his body placement, and even in the physical sense of "putting a feel on a rope" in the way that a person might lift a rope and put some tension on it, but it not actually pulling on the horse...
That is not an analytical or intuitive approach. That is attention to detail.
When I took lessons this summer, the only time we could look at the horse was when practicing at a walk when to give a leg cue. The goal was to develop awareness of what the horse was doing with his body, and which leg was moving when. That was using the conscious, rational part of the mind to teach the subconscious how to interpret feelings from the horse's back to know when to apply a leg cue. With practice, I could stop staring at the horse's shoulders and legs, and use the 'feel' to time cues when they would be effective.
Much of what we say is an intuitive awareness is actually a set of small cues. We can be consciously aware of any one of them, but cannot process all of them fast enough with our conscious mind. However, WITH TRAINING, our subconscious mind can pick up and process that information accurately.
In WW2, folks were trained to identify various enemy aircraft. At first, they would need to compare features and stare at a close plane for a while. With training, a person could get a 1/10th second look at a distance, and accurately 'remember' what it was.
After 4+ years with horses, I'm getting much better at picking up their responses. I don't have to wait until the horse bucks to know something is wrong, and I don't have to stare endlessly at their ears, neck, etc. But this is not an intuitive response to horses, but a trained subconscious review of many physical cues I've learned from observation.
"a very common dressage mental image"
Yes. Mental imagery. I'm sure it helps some. It leaves me cold. I consider a bit to be like a keyboard. It allows me to input information into a computing device (the horse's brain). That mental image might not help many, but it reminds me that a lot depends on how the horse was 'programmed'. I once played around with teaching Lilly to turn based on pressing a finger against the left or right side of her wither. I didn't go too far because she was green-broke and it wasn't fair to confuse her - but she was picking it up easily enough. When riders talk about cuing a horse for cantering, I can't help but think I could train a horse to canter based on the 'cue' of squeezing his withers.
There is no energy going from the horse's rear, bouncing back from the bit and into your hands. If that helps you picture how to handle the reins, fine. I have always granted that Sally Swift's book has helped a lot of people. But I would argue that someone who responds to that mental picture is not a better rider than someone who thinks of a bit as a keyboard (or, to show my age, punch cards).