So I've been curious, whenever I see a new transfer that was approved by the director (or someone else involved in the movie's production like DP or editor), what exactly goes into that? Is it just simply the person glancing at the transfer and going "yep"? I do remember the controversy with William Friedkin over The French Connection Blu-ray, to the point where it was re-released.

I would assume it would depend on the director. The consumer probably would never know. They might just look at clips or caps. If they watch the movie they might see other things they would like to change, so they choose not to watch the entire movie. They probably rely on people they trust, like the DP, to make sure the movie looks the way they filmed it.

The impression I've gotten is that there isn't a consistent answer. Sometimes the director or cinematographer will be extremely hands-on, other times they'll be supplied with a timed master and asked for any comments/revisions, sometimes there'll be an extensive back-and-forth and other times it's assumed one round of feedback is enough, etc.

Color reproduction is subjective. If one were to take an original 1971 print of "The French Connection" and examine the colors, they would not be exactly the same as the colors were back in 1971. The chemicals used in the photographic process do cause color shift over time no matter how well they were preserved. I'm sure you could examine 10 separate preserved prints and they would probably not all match perfectly.

If they're all slightly different, which one is "correct"?

What you saw in the theatre in 1971 was never the perfect reference gold-standard. You have hundreds of prints, running through hundreds of different projectors some of which are calibrated, some not. Some are brighter, some not as bright. Prints were maybe developed in different locations with slightly different chemicals and so on ...

I don't think there really is "perfect" when it comes to color reproduction from old films to the latest 4K (or 8K) film restoration.

Depends on the filmmakers and how demanding they are. Some are fanatic about matching the original answer prints, while others are far more willing to tinker and revise the color correction. Supposedly Cameron spends weeks and months going over his work for approval (and also possibly demands an expensive budget just for the process), while others sign off practically sight unseen.

It reminds me of the whole Dracula on BD issue from 2007. Robert Harris said the BD matched the answer print but others insisted the BD had crushed blacks that obscured VFX work and there was a certain light in one shot that had its color changed, etc. I also recall people insisted the Superbit DVD had better colors than that BD. Heck, 10 years later and some were swearing the Superbit DVD had better colors than the UHD BD.

Whoever did the color correction on the Superbit DVD for Bram Stoker's Dracula did a great job given the limitations of that format. The UHD is very good, far better than the sometimes mediocre job done for BD. Was it absolutely perfect on UHD? I suspect another party could have further tweaked the colors, a few scenes are a little off still but far closer than before.