Has anyone been able to define well or measure differences between vinyl and digital?

Lots more here to read and discover. Give us some time to explore these. 

thespeakerdude

... accept that properly implemented digital, even CD quality, has no sound or so little to be ignored ...

Sorry, no can do. Setting aside the problems and fallacies that are common to typical audio blind testing (this addresses some of them), there have been tests that show hi-res audibility. Here's an example via AES.

... accept that vinyl and other analog formats have a particular sound ... and that we like it because it has those sounds that appeal to us ... figure out what those sounds are and encourage audio companies to work on them and recording and mixing engineers to make better use of them ...

If your claim were true audiophiles wouldn't work so hard to reduce the noise and distortions to which analog is prone. On paper, those specs often aren't the equal of digital, but that's moot if they're near or below the level of audibility.

Read this morning: https://aestheticsforbirds.com/2021/04/07/an-audio-professionals-take-on-vinyl/

Very good summary of this discussion from a layman’s perspective. Covering the two formats and recording chains. I especially appreciate the descriptors used, warmth, richness, and depth. 
If I wasn’t using my mobile right now I’d copy the definitions here to promote a common language for discussion of the differences between these formats.

I can only recommend reading this paper.

The academic research papers are a different kettle of fish.

@cleeds  I included the weasel word little. I linked examples where analog sources were passed through a digital chain of CD quality and the listeners were not able to detect it. I could waste hours on the web finding examples of suitable implemented testing comparing CD quality audio to hi-res audio that would conclude no detectable difference. Can we agree that if there is a difference between CD quality done right and high resolution that the difference is very small, and hard or very hard to detect?

Play any example of vinyl ever made and the closest CD and everyone will be able to tell them apart. Maybe you will find some obscure set where that is not true. Can we say 99.9% of them?

Take the last two paragraphs and put them together. The difference between vinyl and CD is bigger, much bigger than CD and high res.  I am working from the assumption that high resolution digital is good enough to be perfect. Two-four times the bandwidth of CD, 20db or more of added dynamic range and hard to tell the difference from CD. I would say it is near perfect.

I did not say what that particular sound is. Cross-talk was mentioned. No matter what you do, that is there. When you are getting to the inner grooves there is unavoidable distortion. I am not up on the latest in vinyl, but my memory says the best distortion from vinyl, especially at high frequencies is several magnitudes higher than even CD. Maybe it is a combination of the cross-talk and the mastering, and nothing else?  Maybe my turntable setup that I think has a flat frequency response does not?

Every vinyl versus digital argument seems to devolve into an attempt to find some mysterious flaw with digital that cannot be supported with math, engineering, nor experiment. Maybe there is some flaw at CD quality that we can possibly detect. If there is, it is very small. The differences between CD and vinyl are not small. Some progress in understanding would be nice. It is not going to happen by starting with an unjustified conclusion and working back.

My posting issue was not vinyl versus digital except in as much as they are different, and can be use as comparators. Not better. The diversity of situations from recording to playback are very clearly well described in the article from thatspeakerdude that I cited above this morning. I agree that we are looking for progress in this conversation not walls or reverse cycles.

In the playback domain, it is amusing cleeds, that audiophiles, me included work so hard to reduce distortions when each format has distortions of various impacts, some desirable in some situations and others perverse. From tubes to rooms, from vibration to EMI/RFI, the playground is full of hazardous pursuits.

Again though to my OP, I was focused on the warmth, depth and richness as basic identifiable differences between the two formats. Not always, but generally. And we seem to have made some progress in elucidating the circumstances of these differences in the recording and playback chain.

Flaws, like beauty, are in the eye (or the screen) of the beholder.