So I've just read through this whole thread. A few comments.
First, a trivial point. Yes, right --- Harold Faltermeyer. Oops. Anyway....
It strikes me that BDL's tripartite classification makes best sense when conceived pedagogically, i.e. for teaching, especially teaching of relatively beginning students. In a sense, it's not unlike the old wheezes about martial arts masters who force students to wash floors and paint fences and so on, when the students really just want to get cracking on hitting things. The point, of course, is that there are things you learn by washing floors or painting fences that are relevant and important, and furthermore that great beating-people-up skills are dependent on a lot more than learning to hit things as such. Just so, BDL's classification makes the argument that using a knife really well depends on a lot more than just cutting things, including things like board management and sharpening that many people think are irrelevant or a completely separate skill-set. Thus a part of what BDL and KYHeirloomer are arguing about is whether, in fact, board management and sharpening are or are not essential parts of using a knife well.
If the aim of the classification is not strictly pedagogical, however, we come back to my original question: what is it for, in fact? There can never be an objectively superior choice for such classification, because such superiority will always have to do with aims: classification for what purpose?
Rather than muck around with philosophers, let me just put the same point concretely. Suppose I want to categorize the books on my bookshelves. If maximizing space efficiency is the dominant principle, then clearly it will be advantageous to shelve books by height. If maximizing my own ability to find things is dominant, then it is advantageous to shelve books according to a system I have worked out in reference to my usage patterns and habits of thought. If maximizing the ability of someone else to find books is dominant, then it is best to have a system that can readily be imposed upon all books regardless (e.g. Dewey Decimal, Library of Congress, etc.). Now which is the right system? That's the wrong question. Right system for what?
What I was trying to get at in my first post, in my rambling way, was that this "for what?" question hasn't been addressed. I can accept something akin to BDL's system if the question is answered pedagogically, as noted above, but other than that it seems problematic in a number of respects --- as borne out by the discussions here. If the pedagogical aim is matched by the structure of BDL's Cook Food Good writing project, then the question is in a sense pointless: the intended audience consists principally of relative beginners to serious cooking, and they will have an entire volume to become convinced by BDL's arguments on behalf of --- and supported by --- his classification. But what if we're actually interested in a sort of taxonomy of knife skills?
In that case, there are any number of ways to proceed. I'll just sketch two.
In one approach, you start by deciding on a core principle upon which to focus, plus a passably large sample of undeniable examples of the broad class to be examined. Then you begin dividing into piles, differentially, aiming to distinguish among the various examples by a range of monothetic (binary) questions regarding the core principle. By battering at this, trying out as many possibilities as you can, you begin to discern families within the class, and in the same gesture you find certain questions to be of a higher order than others. That is, if I have 6 objects, I find one question that divides them into 4 and 2, then another that divides the 4 into 2 and 2, and then three more questions that split each of the 2's in turn. Because every question relates to the core principle, I can ultimately find a chain of questions that take me from any given example to the highest-order generalities, and I have a workable system.
At base, this is how Linnaeus' taxonomy worked --- which is not the way you were taught it in biology, chances are, because it has subsequently been radically modified by a number of things, most especially evolution and heredity, which were not known to Linnaeus. His core principle was reproduction, which turned out to be a brilliant choice in light of later discoveries.
The disadvantage of a system like this is that it is 100% imposed. It makes no claims to discern any internal logic within the objects classified, but rather imposes a logical system of relations upon them.
The other main way to do things is to take a vast range of potential questions and apply them indiscriminately to every data-point, then look at the enormous chart this produces. This allows you to find groups or clusters that may be useful to analyze on their own terms, but tells you nothing about where the clusters come from.
Another common method attempts to discern, within a kind of combination of the two methods, a common underlying logic that is not imposed but somehow arises from the objects classified. The danger is obvious: one tends to fall into the trap of seizing upon a logical principle that actually comes from the investigator and not the objects, and then seeing the fact that it pans out (and it always will) as confirmation of the system so constructed. Then again, it's worth noting that Darwin did exactly this, recognizing that a combination of heredity, environmental pressure, and really long spans of time would give a reason for Linnaeus' system not imagined by Linnaeus, and he was subsequently borne out by all kinds of things, most notably genetics and paleobiology. So it can work --- but it's rare.
On the whole, the only way to evaluate these different methods and the various systems they can produce is to decide what ends you want the classification to serve. Having done this, you'll find that some methods are more likely to be fruitful than others.
Which all comes around full circle. What's this classification for?