• Originally Posted by Q.G.
"Lots of choices. Pick one." "Define" (in the "pick one" sense). "[...] perceptible by a person"
And still "an objective scale"?

Doesn't work.

The second approach, based on arbitrarily chosen ("let us say") dimensions, does no better.

As you say: "The criteria may be arbitrary"
And there the whole things come falling down. DOF is not an objective thing.

A statement like "The second method is independent of factors such as magnification, viewing distance, photo size, etc." even cannot be a thing related in any way to DOF.
DOF, in essence, is dependent on factors such as magnification, [etc.]
Technically correct, but not useful as a practical matter. OTO CoC are useful as a practical matter (caveat below), as in for a given format we all know what we are talking about, and therefore can actually "pick one". If someone picks another for the same criteria, one can extrapolate with ease and the subject becomes somewhat objective, at least enough for practical discussion.

I fully agree that none of the other things are related to DoF in the least, only to "apparent" DoF which is purely subjective.
Field of View is a function of format if other factors are the same, and if those factors are the same the actual properties of DoF will be identical. The subjective effect might be quite different however, say between 100mm on minox vs 8x10 inches. The minox frame will offer the same properties within a severely cropped area and appear as a compressed telephoto lens. The 8x10 will be very wide and offer the perspective of and focus characteristic of a very wide focal length. This is because the perspective of a focal length is relative to the format. Folks tend to forget this. To offer the same FoV between formats one must change the lens or distance and focus, and you are no longer comparing anything meaningful concerning DoF, not that it's overly meaningful the other way either.

The CoC for a given format ,focus, focal length, and stop allows one to calculate DoF in a practical way (there's that word again) I seldom calculate for stills because I can see it, but for motion picture where things are very fluid and judgments by eye can be very difficult I use a thing called a SamCine calculator. Basically a slide rule that allows you to calculate DoF for different format/focal/t-stop combinations using CoC as the objective criteria. Understanding the CoC implications for a chosen format is important to using the device. Where the CoC remains the same it will not have the same visual effect as a practical matter across formats, regardless of your subjectivity or objectivity, therefore CoC alone doesn't tell you anything. It's more than just about subjective/objective, it is about practical application, and there isn't some magic bullet formula that you can carry in your pocket that applies across the board. A only a working knowledge and experience can really inform. In other words, get out and shoot. A couple of exposures are worth more than all the words in this thread, mine included.