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

@johnread57, this is not at all about vinyl, but about analog tape recording from the perspective of a recording engineer. There are some great sound bites to listen to. Summary: Digital comes out just like the way you put it in. Analog recording does not sound like what you put in. It may sound better. It may sound worse. This guy has 9 albums nominated for Grammy’s. He is not a hack.  Vinyl has a lot more distortion than digital and it rises with frequency. Add this to the qualities of vinyl that others has posted here.

http://recordinghacks.com/2013/01/26/analog-tape-vs-digital/

Here is another from a musician (60 recorded works) and recording engineer (100 albums)

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

There is MOFI, and this, http://drewdaniels.com/audible.pdf

The test results show that the CD-quality A/D/A loop was undetectable at normal-to-loud listening levels, by any of the subjects, on any of the playback systems.

Step 1, accept that properly implemented digital, even CD quality, has no sound or so little to be ignored. Step 2, accept that vinyl and other analog formats have a particular sound, or many sounds, and that we like it because it has those sounds that appeal to us, as living, breathing humans. Step 3, 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. Step 4, accept that we are mostly a group of old buggers, and maybe young people don’t like the same thing. I tried to research more on this last item but I didn’t find a lot of work on it.

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.