I always thought the term flat field meant the focal plane across the visual field was in focus from edge to edge, whereas a curved field meant that some % of the periphery of the focal plane was not in focus when the center was, but could be brought to focus, by re-adjusting the focus wheel, so the plane of focus was curved, by design.
.... It is not yet clear to me whether curvature of the focal plane and barrel or pincushion distortion of a grid are inextricably linked when it comes to binoculars. In other words, can one have a relatively flat focal plane in a visual field, combined with barrel or pincushion, especially with multiple glass elements?
.... So, what does flat field refer to then when it comes to binoculars? A curved or flat focal plane, or a visual distortion that is seen when looking at a grid? (or both)
The term "flat field" is thrown around a lot; some use it to mean "sharp to the edge", other times people use it to mean "straight lines (no pincushion)." But my understanding is that a "flat field" most accurately refers to field curvature, as you noted.
I've read enough of Henry's posts to know that correction of field curvature / astigmatism (which impact sharpness as you move to the edge) is NOT related to rectilinear distortion (barrel vs pincushion).
I have a feeling these concepts have become linked on binocular forums in no small part because of the "Swarovision" prominence and dominance. When Swaro introduced the EL SV series, all the talk was "flat field! flat field!" and these binoculars happen to have excellent correction of astigmatism + field curvature (ergo very sharp to the edge) and ALSO they chose to aggressively correct pincushion distortion so they have high AMD (rolling ball effect).
So I don't think it's "inextricably linked", other than the colloquial linkage that has arisen as a consequence of the Swarovision defining the "flat field" paradigm for the modern roof. So in many people's minds, thanks to the EL SV, the term "flat field" makes them think "sharp edge to edge, straight lines, but also rolling ball".
To me the impression of 'flatness' in an image has more to do with depth of field, and value structure, rather than relying on an engineered spatial distortion across the entire field, introduced by optics.
As above, I think it's important to distinguish between "technical" terms (low field curvature / pincushion vs barrel distortion / etc) that are precisely defined, vs. colloquial subjective descriptors. If someone compares two binoculars and comments "wow the image on Binocular X feels so much deeper, whereas Y just feels flat in comparison" ... well, there's a lot of things they could mean by that!
I currently have in my possession a 10x32 EDG, an 8x42 Monarch HG, a 10x42 Trinovid BR, and an 8x32 Ultravid HD, listed in order from "flattest" to "most curvalicious". I absolutely notice what Tobias described as "images render space in a flat, compressed way" with both Nikons comparing to the two Leicas. Something about the view in the Nikons (which both have "field flatteners") makes the depth of the view feel a bit "compressed"; I am very confident it's not my imagination... as you noted, with photography you can create a sense of artificial deepening or compression by manipulating the visual plane / focus depth, so it's a real phenomenon in theory.
If I had to describe it, my best analogy is a "diorama effect", where layers of depth feel flattened as though real objects appear to almost be flat cut-outs vs 3D. For example, looking over the tops of trees, where there's a lot of depth from the closest to the farthest tree, the full depth of the trees feels compressed/flattened towards the viewer, and the trees themselves don't present as "deep". The EDG (which most aggressively corrects pincushion + field curvature, although neither as much as a Swaro SV) shows it the most, the MHG to a lesser extent. The UVHD 32, which has plenty of pincushion AND field curvature with zero AMD, feels incredibly "deep" and "3D" in comparison when I swap. Now, I realize the 8x mag of the UV gives it a "depth" advantage vs the 10x of the EDG, but since I have two other models (one 8x and one 10x) with which to compare, I do think there's something to it.
As an interesting side anecdote here -- the 8x32 UVHD are my wife's binoculars. She's a one binocular gal, and she LOVES those Leicas, and only uses them. When I get a new binocular in house to play with, I like to hand it to her to try out just to get her off-the-cuff impressions (she doesn't know anything about technical optical stuff, just raw subjective reaction).
I was really hoping she would like the EDG, since she had used Nikons for decades prior to getting the Leica this year, and more importantly the EDG have a very narrow minimum IPD, and my wife is on the extreme narrow end (52-53mm range). When she tried the EDG, however, she immediately said, "oh, these seem weird! they are very clear but it feels like I'm looking at a microscope slide!" (she is a lab scientist so this wasn't a random analogy). She switched back to her Leicas and said "yes, this feels so much more natural!" When I pointed out the 10x vs 8x focal plane difference, she grabbed the 10x42 Trinovids and compared and she definitely felt it was something unique to the EDG, not 10x magnification (and the Trinovid has a smaller FOV than the EDG, so it's also not the 8x UV's advantage in FOV).
After probing a bit to understand what she meant (I tried my best not to "lead" her by revealing anything about flat fields vs curvature etc) it was clear she was experiencing the phenomenon noted above, and her reaction of "like looking at a microscope slide" was her trying to convey that it felt like the image was "compressed" and felt "too close" to her, like a picture being shoved in front of her face, whereas with the Leica it felt more "natural" as though she saw the extra detail, but the subject still felt "properly placed" in space.
So while she could recognize the quality of the EDG optics, she just found them subjectively distasteful and they were her least favorite of the 4 options noted above.
EDIT: I also wanted to note that, while I do NOTICE the effect, it doesn't really bother me too much. The EDG view is otherwise just phenomenal, and thankfully they implemented a very "tasteful" approach to flattening, retaining just enough pincushion and field curvature to not make it feel excessively flat in real world use (e.g. panning doesn't feel unnatural, in fact the 8x42 MHG to my eyes exhibits more "rolling ball" than the 10x32 EDG).