The institute was not called
The Inquiry of the Movement and its Improvement.
This is an indisputable fact.

The order of the words is not insignificant.
If the purpose is awareness, it comes first.
If the purpose is movement, the order is reversed.
These formulations do not mean the same thing.
In inquiry, attention to detail is decisive.

Chapter 4 in the ATM book is titled Structure and Function.
A new translation establishes the correct title as The Structure and Virtues of Its Function.

This is Moshe Feldenkrais reasoning to explain his discovery and what later would become a profession. I have a notion that many are unaware of the detailed theoretical explanations given already in the late sixties.

From this chapter the English name Awareness Through Movement is somehow created - I can not find it in the text.

"...In the beginning of 1982, Moshe Feldenkrais returned home to Tel Aviv, shortly after undergoing a major skull operation for a subdural hematoma.
The operation had been successfully performed in Switzerland, and Feldenkrais had remained there for an appropriate convalescence. Now that he was at home, he wanted to regain the physical ability to work on people giving Functional Integration sessions as quickly as possible.
I worked on him three times a week, giving him Functional Integration sessions. What struck me immediately was a deterioration of the relationship between head and trunk. I knew that wonderfully organized neck from before the operation, and Moshe's old ability to move his head with the greatest of ease in all possible directions..."

see Panel 3

A text from Acquiring the Feldenkrais Profession
Structure and function, page 34

"We can define four relevant factors that constitute a human being acting within his or her environment:
the skeleton, the muscular system, the central nervous system, and the environment.

Each of these require the special professions that can deal with the respective issue: orthopedists, surgeons, neurologists - psychiatrists, and professionals who deal with the various aspects of the environment - constructors, architects, carpenters, and the like. We can envisage the four factors as the four sides of a quadrangle:"

 "*The corner 'environment-CNS' symbolizes the interface through which sensory information arrives from the sense-organs to the brain and serve there as the basis for planning suitable actions, either to change parts of the environment, or to adjust to it. The Feldenkrais Method deals with efficiency and with the possible alternatives for those processes as well. 

*The corner 'CNS-muscles' denotes the processes that come into play with the intentionality of the action itself: deciding to take the action or choosing a non-habitual way of acting, etc., again, typical considerations of the Method. "

"*The corner 'muscles-skeleton' refers to the conversion of muscular effort into movement, and all the considerations of alternative options and efficiency are our concern.

*Finally, the corner 'skeleton-environment' alludes to a twofold interaction: adjustment to the environment, including the tendency to look for support and the various anti-gravitational responses, and the actions by which one does work on parts of the environment, by exchange of energy. Here again, the Method has its own way of clarifying the situation and seeking efficiency."

Professor Karl H. Pribram, with a broad background that includes neurosurgery, psychology and brain science visited the San Francisco training. Talks between him, and Moshe Feldenkrais are published.

Classical Feldenkrais uses Image of Action
Karl H. Pribram uses Image of Achievement

Karl H. Pribram published a book titled Brain and Perception, with the subtitle Holonomy and Structure in Figural Procession.

The book is a collection of manuscripts and reported studies first published in 1991. A later edition was published in 2011. The Feldenkrais Foundation is listed among the institutions that provided a grant for the publication.

Chapter six, titled Images of Achievement and Action Spaces: Somatic Processes in the Control of Action, contains material relevant to the distinction between action and movement.

What attracts my attention is that he writes that the distinction between movement and action is often confused in scientific literature. As so often, different professions use different designations. His example is how ethnologists use the term behavior to describe a sequence of movements. In experimental psychology, a behavior is the act, the environmental consequence of movement. He states that images of achievement is a construction as other precepts. Some achievements are to a great extent genetically programmed as those developments leading to walking and eating, swimming and talking intermediate, and achievements such as writing are for the most part learned.

Skills such as riding a bicycle, skiing, golfing, musical performances, and writing all depend for their execution on the development of appropriate images of achievement. It is characteristic of such skills for achievement to be essentially invariant across movement. Entirely different movements carry out writing on a pad of paper than those that carry out writing on a blackboard. Nor is writing dependent on using familiar musculature.

Pribram gives a clear and thoughtful example when a right-handed person walks down to the beach and write the personal signature in the sand with the left big toe. The result will be both readable and identifiable as the signature. Another example is placing a piece of chalk between the teeth writing on a blackboard.

The signature is imaged - consciously or unconsciously; different sorts of movements are equivalent in producing the imaged consequence, the act.

To my understanding this reasoning corresponds well to classical Feldenkrais, that is:

Addressing the elements of action to
enrich the self-image
and, as a consequence also,
the image of action (achievement),
all together,
as an undivided whole.