To further enrich the debate, I’ll just add the translated opinions of some German mappers in this thread:
And here’s a summary of the definitions of these surfaces on the English OSM wiki over time, juxtaposed with tag usage history.
The bottom timeline uses codes to refer to the top images and the descriptions in the middle. So 21A means images 2 and 1 were used simultaneously, with 2 first followed by 1, together with description A. The purple horizontal line represents half of all usages of cobblestone.
Considering the 218k OSM objects with surface=cobblestone (162k) or surface=sett (56k), it seems to me that:
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~25% are surface=sett that in reality have a flat top (56k out of 218k)
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~25% are surface=cobblestone that in reality have an arched top (whenever the image for cobblestone was 6 or 9)
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~50% are surface=cobblestone than in reality may have an arched or a flat top
Remarkably, usage of sett has accelerated by the time that the aforementioned thread in the German forum started getting replies.
Considering both vernacular and technical usage, maybe it would be most useful in the long run to define these surfaces like this:
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paving_stones: blocks or stones with smooth flat top, tight gaps, images 1 and 2
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sett: hewn stones with flat top, wide filled gaps, comfortable to cycle and walk on, uncomfortable on high heels, images 3 and 4
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cobblestone: hewn stones with slightly arched top, wide filled gaps, uncomfortable to cycle on, difficult on high heels, images 5, 7 and 8
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cobblestone:raised: natural or hewn stones with very round/irregular top, wide empty gaps, fixed to a bedding, difficult to cycle and walk on, uncomfortable to drive on, images 6 and 9
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cobblestone:flattened: never
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pebblestone (for contrast with cobblestone:raised): loose natural small rounded stones
Back to my first post in this thread, this would make the Portuguese pavement and the polyhedral paving a type of sett most of the time. Portuguese pavement does resemble image 3 and is uncomfortable on high heels (as on sett), and crazy paving does resemble image 4 except for the regularity of stones when viewed from above (also uncomfortable on high heels according to multiple anecdotal reports on the Internet). Crazy paving would sometimes be sett, sometimes paving stones (like flagstones would), depending on how wide the stones are in relation to gap sizes.
2 would be called Block Paving in England, today. They are used for footway crossovers (“dropped kerbs”) and personal hard standings, and occasionally for low traffic residential roads, more likely private ones.
Paving stones is generally only used for pedestrian areas with large, rectangular, lumps of concrete. These will eventually crack if even just domestic cars drive over them, especially if the vehicle is turning, so are never used for intentional vehicular areas. When reporting ones that are broken (usually by abuse by motor vehicles, my council just calls them “slabs”.
It was the original image, and there was no visible consensus to change it.
I also believe that using a relatively extreme surface like your second image to illustrate “cobblestone” would not do a good job at representing what the value is actually used for in the database. Using the first and third image alongside each other would be ok from that point of view, though.
In an effort to get away from discussing things people did or said in the past, though, and ignoring the existing values for a moment, here’s how I would intuitively group your example images:
Group A (probably relatively uncontroversial to call it “paving stone”):
Group B:
Group C:
I’m glad our intuitive understanding is quite the same. Also, this understanding seems to be in agreement with those English Wikipedia articles on sett and cobblestone.
If we were to adopt those definitions in OSM, then I think it would be interesting to use images 3 and 6 for setts and images 7 and 9 for cobblestones to reduce the number of borderline cases, thus making edit wars less likely.
But then we’re brought back to the question of whether OSM should always stick to the status quo.
To contribute to the discussion, this is what StreetComplete shows for paving stones, sett and cobblestone, respectively. I think it fits quite well in the groups presented by Tordanik, so this is really just FYI:
Pretty much.
OSM definitely shouldn’t be stuck with the status quo if there’s a clearly better alternative. But in any such situation, we need a plan to get from “here” to “there”. What I’d like to avoid, if possible, is a re-write of the wiki definition such that previously correctly mapped roads become incorrect overnight.
Perhaps it’s worth looking at an example of what I’d consider a successful transition: The subdivision of “farm” into “farmland” and “farmyard”. What made it work, imo, was that there were two new terms. This meant that landuse=farm wasn’t instantly wrong – merely imprecise – and that you could see which tags were following the new definition and which ones were left over from before the change and needed a revisit.
Based on that model, the cleanest solution for our current problem may indeed be to not just promote one more precise value (sett), but two of them. If we had a less ambiguous synonym for the technical definition of “cobblestone”, that would not only allow us to pull off something similar to the farm subdivision, it would also take care of the misunderstandings between technical vs. colloquial usage.
Of course, there may simply not be a suitable English word… 
I rather like the systematic deconstruction of our terminology here. I don’t know if it will resolve anything but it’s very interesting, and illustrative of many other areas of tagging.
@westnordost: your example of paving stones looks awfully like brick/block pavers to me (hadw explains the differences very well).
Block pavers and bricks would be mapped as paving_stones in OSM, so I added this information to the wiki.
I think that flagstones would be mapped as paving_stones too. What do you think?
I could have sworn I’d used a brick or block paver tag value in the past. Seriously they do need to be separated out from paving stones: they are a very common highway surface on service roads & lightly used parts of residential streets.
Flagstones on the other hand clearly are paving stones. I presume they are the original paving stone.
+1 for block pavers not being “paving stones”.
Some people have used the somewhat rare “bricks” and “brick” values. No other value currently used seems to refer to block pavers.
Could it be that you want it separated from paving_stones because of the differences (such as slipperiness) between natural stones and artificial blocks? Or would there be another reason? Would there be cases, such as artificial blocks made to resemble natural stones, that would confuse mappers?
Would block pavers and bricks need to be separated as well? For what reason?
The surface tag is a little bit problematic because it is, by definition, an “open set” (it can have really MANY values). For data consumers (applications), it is better to keep the number of values tractably low. For users in general, it is better if apps can make use of mappers’ work.
Paving stones, as the term is used in England, are concrete. The key thing that distinguishes them from block paves is size. They are large enough for vehicles to create sufficient shearing moments to break them.
Err, back to the cobblestone topic:
Just my thought! I previously proposed “natural_cobblestone” for those uneven, unhewn, non-sett cobblestones. It’s not a real English word, but it should be quite clear what is meant, should it not? Another alternative: “unhewn_cobblestone”
I thought I’d posted this before, but must be mistaken.
If we’re just looking for a synonym to cobblestone to facilitate a transition, how about surface=cobbles? Or, if we want to get silly, surface=lumpy?
We are not looking for a synonym for the colloquial “cobblestone” but specifically this cobblestone made from natural unhewn stone. The clear distinction to cobblestone as generic term should be reflected in the name.
The problem with the word natural is that it might be understood as referring to substance, shape, or both. Cobble already implies substance, and we’re trying to identify shape, so maybe unhewn would be interesting (for example, unhewn_cobble or unhewn_cobblestone). It is not very well known though, judging from Google Images’ results.
Also, cobblestone is generally understood as having some binder (cement, mortar) holding the stones in place. Using cobble alone might be understood as loose stones.
In my dialect, cobblestone and cobble (singular) would imply a single, possibly unbound, item whereas cobbles and cobblestones would refer to a collection of them, possibly bound, possibly not. In my dialect it would be natural to refer to a cobbled street and peculiar to refer to a cobblestoned street. In my dialect, cobble implies substance, shape.and usage. Cobblestone doesn’t imply usage as strongly. “A street of cobbles is constructed using cobblestones” seems more natural (to me) than “A street of cobblestones is constructed using cobbles.”
Other dialects of English may differ on some or all of those points. Which may not be too important if we’re just trying to come up with a new term so we can deprecate the abuse of the old term. It’s not ideal, but one has to expect rough edges. 
So, unhewn beats natural and cobblestone beats cobble/cobbles.
So, how about unhewn_cobblestone then?
Glad to see some good suggestions, and at least to my non-native ears, unhewn_cobblestone sounds like a workable option that gets the distinction across.