What is the best way to appeal an unfair qualifier exam decision?
It is hard to know the exact issues for an oral exam we were not present for and for which we are hearing only the candidate's version.  Based on your own description it sounds like you feel very strongly that you know the material flawlessly and were treated unfairly.  I'm sorry about that.  However your description did not convince me that that's what actually happened: rather it sounds like the committee had some issues with your performance that you don't yet fully understand.  Since you are allowed to retake the exam, you should find out as precisely as you can what went wrong and what they want you to do better.


  I was well prepared for all of my exams


Isn't that for the committee to decide?


  and did excellently on exams B and C. 


May I ask: is that the feedback you got on exams B and C: that you did excellently?  Or was it just that you passed?  It matters: if in two out of three exams that committee really felt that you did "excellently", then they think you are an excellent student and there will be a lot of support for you to stay in the program.  Also, do you have the same examiners for all three exams?   


  During the oral part of the exam...The textbook I was being tested on used Z, as did all 3 of the courses I've had on subfield A.... I asked whether the terminology issue was the deciding factor and it apparently was not. 


You gave a very long description of something that took "five minutes" of a one hour exam and was explicitly said not to be a cause of your failure.  You are focusing very strongly on whether it was reasonable for you to know the terminology.  But then afterwards you just looked it up on the internet and decided that it was not something you need to know without talking to anyone else about it.  That's a problem.  Another observation is that -- and this is something that happens to lots of students - you let your entire performance get derailed by an inessential minor point. 


  For the sake of identifying what I did wrong, I spoke with one of the professors who administered my exam.


You should speak to every professor who administered the exam.  Exams are not failed because one of the committee members didn't like your performance: at the very least the majority felt that way, and much more commonly they all did. 


  The most I can make of this is that the terminology issue made him and the other committee member think I'm an idiot, and then they were much less fair to me from that point on.


I wasn't there, but I think this is not a very good guess.   Since there seems to be a lack of clarity about the result even after speaking to one of the examiners, I would consider asking for a written evaluation of your performance.  It would be a good idea to discuss this with your adviser first and make sure that this will be properly viewed.  I think it is a very reasonable request: you passed two out of three exams and will take the third one again.  You really want to know clearly -- and spend time thinking about and taking into account -- the reasons for the failure.  


  . I used one version of Y to solve this problem, and he complained that (though correct) how I did this demonstrates that I don't know what I'm doing. 


I don't understand what that means.  As you've told it, it certainly sounds annoying.  But he must mean something by it.  Your conclusion


  It seems that he'll complain regardless of how I do things.


is the most negative and unhelpful possible one.  Does this faculty member have a reputation in the department for being so unreasonable?  

You mention that your adviser was there for the deliberations [but I guess not for the exam itself?].  That should be extremely useful: you can get your adviser to explain their decision to you.  If he does not understand why you failed despite being at the deliberation, then you do have a problem.  If that is the case, definitely bring up the prospect of getting a written decision that you can then go over with him.


  They did suggest that I might have been rude, but I don't think I was.


Whoa.  Your advisor thinks that you may have been rude to the exam members, but you have just brushed that off because the one exam member you spoke to didn't specifically mention it?  

I'll give you the honesty of a lifelong academic who is a complete stranger to you: to me, you do sound a bit rude and disrespectful.  You look for confirmation on the internet that you are getting asked silly questions.  You say "In this case, the one professor I've spoken to is definitely wrong."  And you write


  There is nothing to study. The professor I spoke with's claims about my knowledge are contradicted by their own statements as well as my performance on another of the exams (which builds upon knowledge of field A). 


To hear a student who has failed an exam say that there is nothing to study the next time around is a huge red flag.  Maybe you know the material well according to your own standards and previous experience.  But if you know it perfectly: you are in the wrong place.  No one who has nothing to study belongs in academia: you already know everything, so you have nothing to learn.  Also you happen to know that your examiners are wrong, do not understand the material as generally and intuitively as you, and cannot even make logically non-contradictory statements in explaining themselves.  Well, look: either you're right or wrong about this.  But if you truly feel this way, then either way, at the very least this PhD program is not for you.