"In what way?"
We'll, it is obvious that we disagree in a number of ways.
At the risk of repeating myself, I've given numerous examples of successful artists who deliberately set out with an intent. When they synergistically interact with others exploring a similar intent, it is called a "movement".
I find your definitions to be a convenience of argument rather than supported by any historical reference. "Picasso and Chagall were very intuitive with definitions coming later", intellectually sounds good, yet both artists had very definite ideas going in. Theories which were arrive at by many means, including revelations informed by exposure to other well formed ideas (no man is an island), then they went on to some form of organization of those new thoughts in order to put them into consistent action. Whether they were "labeled" yet is irrelevant to the initial formation of the idea to be explored.
Again, as an example, Cubism was an Avant-Garde "invention" regarding notions of Time and Space, where objects were studied, disassembled and reorganized to depict the object in a greater context than had been visually thought of before. Oddly, it is thought that Picasso's deliberate efforts (intent) to depict the human body with greater solidity is what led to its' disassembly … and the subsequent mindful act of freeing himself of conventional rules of perspective and notions of dimensional space.
The above is why I personally think "Intent" is a very appropriate word in the context of Art. Intent suggests wiggle room for discoveries (rational or intuitive, from the artist themselves, or from other artists, even those from other non-visual art disciplines) to redefine, refine, and evolve the initial intention of the artist. "I intended to do this, but in reality I discovered/realized/learned about this during the process, so now that is my intention." The word has an implied "fluidity".
I find the statement that "Artists don't understand their own work", to be curious at best. I think Cindy Sherman understands exactly what she is doing and why … which is supported by a very consistent body of work. It is not necessarily the duty of the artist to make the viewer understand. Duchamp's fusion of Cubism and Futurism (Nude Descending a Staircase), requires some knowledge of Cubism and Futurism, and an intelligence on the part of the viewer regarding what he was doing by combining two notions … but he knew exactly what he was doing, and why. This is not to say there are not artists that don't understand their own work … but they are usually unsuccessful due to that very reason. My interest lies in learning from those who succeeded.
Contrary to your assertion that conceptual work is "not particularly rewarding for the artist or the viewer", my experience has been the opposite. Amongst other more conceptual artistic approaches I'm interested in, I find Neo-Dada Fluxust Art to be a highly rewarding on many levels … but freely admit that it can be an acquired taste with a "Manifesto" (i.e., "Intent") that can take some time and effort to come to understand before you "get it". More well known partitioners include Joseph Beuys and Yoko Ono. Fluxus is attributed with redefining what art can be, and as a movement had, and continues to have, an impact on visual thinking regardless of whether anyone "likes" it in the traditionally narrow sense of the word.
In fact, while there are a number of photographic images that I've reacted to on an initial gut level, some became more deeply moving to me once I more fully grasped what was in play (sort of peeling the opinion to reveal more and more layers) … where others simple were gut reactions that faded with minute familiarity … like cutting open a hollow peach … lusciously beautiful, yet empty.
I vividly recall the onions, and swiftly forget the "hollow peaches".
IMO, Photography as Art is simply a medium that has been disciplined to an artistic intent … not an art unto itself.
I know this flies in the face of what you personally believe, and obviously that's where we will probably continue to disagree … but you asked, and I answered as best I could.
- Marc