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

I've seen accounts of experimental results seemingly proving that CD format is perceptually transparent. Also, accounts of experiments seemingly proving the opposite.

Not satisfied with what I've seen in these papers, I conducted my own experiments over the past decade, where I controlled the whole recording and reproduction chain. The results, for me personally, were more definitive.

I have to disclose at that point that I'm educated, certified, and worked in the areas of Physics and Neurophysiology, including stints in several national labs, in Europe and USA.

So, my standard of inquiry is rather high. When the outcome of an experiment depends on accurate detection of one photon vs two arriving within a given microsecond, it has to be.

Experiments in psychoacoustics are hard, mostly because music signals and the final "instrument" - human hearing system - are very variable. Thus, I put more faith in meta-studies rather than in individual experiments, including my own.

Meta-studies, for instance like the one I already referred to in this thread:

Proponents of this or that point of view like focusing on one particular study, or a handful of them, proving their point. I don't believe that's the way to go.

For instance, the oft-cited Boston Audio Society study (BAS study), which seemingly proved the perceptual transparency of the 16/44.1 digital loop, didn't conduct one of the mandatory steps any experimental science professional would do - calibration.

If an experiment is to elucidate the importance of distortions, one absolutely has to take into consideration the nature and levels of distortions inherent in the gear involved in the experiment. What do we have in the case of the BAS study?

https://www.bostonaudiosociety.org/explanation.htm

"The Principal System

The playback equipment in this system consisted of an Adcom GTP-450 preamp and a Carver M1.5t power amplifier."

What is Carver M1.5t power amplifier?

SINAD reveals that the amp's distortions are high and rising with frequency. "... distortion-free range = 14 to 11 bits". "Distortions rises @ 66 watts". "Max power = 273 watts @ 44 dB SINAD".

What the designers of the study should have done could include at least, back in mid-2000s: taking the Principal System into a certified anechoic chamber, and measuring its performance, including its distortion profile, using certified calibrated instruments.

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.