I understand generally interval tunings can be perceived through the "beating" phenomena. But that seems to be something that can only be heard in a testing context. Like playing two sine waves in a lab. In a real music performance I imagine it's difficult to hear beating.

To compare tunings used in real music performance I'm using this page with 4 different records of Bach's BWV 864 in 4 different tunings: 
<https://en.wikipedia.org/wiki/Musical_tuning#Systems_for_the_twelve-note_chromatic_scale>.

Upon first listen I don't hear a difference in these recordings.

After several listenings I sort of think there is a difference in what I perceive as timbre. Some seem a bit softer than others, but it's *very* subtle.

This leads me to a few personal observations:

My piano (which I'm sure is never tuned well except a few weeks after a tuning) has a pleasing tone to my ears in sharp/flat keys but kind of harsh tone in the basic keys like C and G major. Perhaps this isn't tone/timbre but rather tuning.

On my guitar I can carefully get open D in tune and then hear that open C sounds horrible and I have to shift everything around until I get all the open chords sounding decent. I sometime attribute that to good tuning for the octave in open D becoming an out of tune 6th in open C and visa-versa. But it could be bad intonation on my cheap guitar. Anyway, it's possible I'm hearing the tuning problem.

A final observation: none of the 4 Bach recordings above is 'out of tune' unless one is fanatical about a particular tuning. 

I don't really know if I *hear* a difference with these various tunings. Maybe I do, but I mistakenly perceive as a very subtle timbre difference.

How do professional musicians (with better ears than mine) perceive it?

How many musicians could identify those 4 tunings in a blind comparison?