Hi Charlie! You've pretty well answered your own question, I think, but I'll throw in a few thoughts.
It's a great question you ask, and the issue you raise is a source of a lot of misunderstanding, in my opinion. On on hand, naturally, when you seek the services of an expert in any field, you generally do it because you believe that he or she has expertise you may lack, and you trust his opinion and judgment. In seeking an expert, pro, instructor, coach, ... carpenter, plumber, doctor, accountant... you put yourself largely in their hands, trusting them to identify and solve your problems for you. You go to them because you need them. It would seem ironic, then, for that expert to ask you "what do you need?"
And yet, of course, they must. What the expert needs to know, before he or she can help you identify your specific needs, is why you sought his services in the first place. As you suggest, they need to know "why are you here?" What's the problem? What do you want? You don't hire a carpenter or an architect and simply say, "make me a building." Are you looking for a home? A shopping mall? A hospital? An outhouse? Obviously, you do have needs that the expert cannot know about until you communicate them to him. Your needs--your reason for taking the lesson--may be specific, or they may be vague. But only if you truly don't care whether you get a doghouse or a prison or a hotel can you simply say, "make me a building."
It is a partnership. You need to tell the instructor as much as you can about what you want, or think you want. Then the instructor can do his job and figure out the best way to help you get there. Otherwise, you must take whatever you get. Yes, that can be fun, too--Weems has often recounted his story of going out with an instructor and simply saying, "give me your best shot." But that really only works if you honestly, truly, don't care what you get, have not identified any specific goals, have no hidden agenda, and perhaps come with a highly developed ability to self-assess and self-coach. It would be entirely unfair to that instructor--and a waste of your own money and time--to finish that lesson and say (or even think), "I did not get what I wanted." If you don't identify any goals, wants, needs, or desires, then whatever you get in that lesson, whatever you do, wherever you ski, whatever drills or focus you work on, whatever terrain, speed, conditions, or park features you end up on, ... you got what you asked for.
It's obvious from the discussions at EpicSki and elsewhere that the question of "becoming the best skier I can be" will entail different answers for different people. What is "good" (not to mention, "best") skiing in the first place? Who are the "best skiers"? Are they racers? Mogul skiers? "Big mountain" or "extreme" competitors? Are they movie stars? Are they freestylers? Free-heelers? Aerialists? Are they specialists in any of these things, or generalists who can do them all to some degree? Simply asking yourself about what YOU think a "good" or "best" skier is would be a good start to giving the instructor the information he needs to help you become the best you can be. (And even if you think being a versatile "generalist" is the ideal, you can't work on everything all at once. What--if anything--would you like to focus on today? You might want to leave that up to the instructor--but you might not, too. Tell 'em, either way!)
Of course, I suspect that you will have already answered this question of what is "the best you can be" to some extent, simply by choosing an instructor who "knows you" and who you feel shares your ideals about skiing. But that is different. If the instructor already knows what you want, obviously, you can skip a lot of the "getting to know you" part of the lesson. If your goals and motivations are already "known," then you certainly can "depend on [the instructor's] expertise to determine what you need to work on," as you suggest.
Likewise, in some situations, some goals are more or less implicit already. Racing, for example. If you're a racer, or you've signed up for a "race camp," it's at least likely that one of your motivations is to get faster in the race course. While even that is somewhat of a generalization, and there is room for specific details and individual motivations within that larger goal (and a coach would be remiss not to inquire), a race coach is probably justified in making a few assumptions about why you are there, and what will make you happy.
However, you have spoken of skiing with "an experienced coach totally familiar with one's skiing strength and weaknesses," and I urge caution even there. It is actually not your "skiing strengths and weaknesses" that the coach or instructor is asking about when he asks "what do you want to work on?" A good instructor will become very familiar with the technical aspects of your skiing in extremely short order--sometimes within a turn or two. What the instructor needs to know is what he cannot see: what are your goals? What motivates you? What do you love about skiing? When you think of becoming "the best you can be," what image of "good skiing" comes to your mind? It's not asking a lot, and it certainly isn't an attempt to cop out or shirk the responsibility of identifying what you will need to further your progress. Like "taste," motivations vary widely. "Progress" assumes--and is measured by--a goal or destination. Until you know where you want to go, it's impossible to identify the right direction to get there.
On the other hand, it may well be the journey that you seek, not the destination. When that is the case--as it likely was for Weems in the "give me your best shot" scenario--you may well enjoy the adventure of being taken somewhere unknown, unexpected, and unplanned. Nothing wrong with that--and good instructors are experienced and effective at taking you into uncharted territory, physically, mentally, and emotionally, when that is your true motivation. But "the journey" is not the same as your stated goal (destination) to "become the best you can be."
Either way, it is always true that the more you can let your instructor know about what you want, the more effectively that instructor can help you progress. Don't worry if your goals are vague or specific--just tell the instructor either way. Give 'em whatever you've got. Good instructors are highly skilled at turning vague goals into specific objectives that are relevant to those goals. They can help you clarify your goals if they are vague. But you've got to help them out--it's in both of your best interests.
No matter what some people may claim, I really do not believe that they have absolutely no personal goals, hopes, desires, questions, or motivations, and that they are willing to leave what they "should want" entirely up to me. There is some reason you paid for this lesson. What can you tell me about that reason?
The most unproductive--and unfair--thing you can do is keep your instructor guessing at what you want, and whether or not you're getting it. Please do not wait until the end of the lesson to say, "you know, that was very interesting, and I feel that you've really helped me improve my edging and carving skills, but I had really hoped we'd slide some rails and get into the half pipe today...."
So that's what they're asking when they say "what do you want to work on today?" What are your goals? What floats your boat? What's on your mind--today? They're not looking for what exercises or drills you think you should practice--unless that truly is what motivates you for some reason (in which case, of course, that would be exactly what they are looking for, although you could expect a sincere "why do you want to work on that?" in response.) Indeed, the more specific your reply to "what do you want to work on?" the more likely you'll face more questions about why you want to work on that. Kim Peterson, who authored PSIA-RM's "Guest-Centered Teaching(TM)" model, has famously observed that "better technique" really is not a motivation that brings people to lessons, even if they say otherwise. Technique has no purpose in itself. What really motivates us is what that technique will do for us--whether that is win races, make us less tired or more confident or more elegant, help us enjoy more challenging terrain, feel new sensations, attract the opposite sex, become rich (yeah, right), make us more fit, keep us safer, or just generally "have more fun" (what is fun for you?).
Most students haven't really given this question much thought, but they should. If you really want to leave it all up to your instructor, that is your choice. And once you convince your instructor that that is truly what you want, a good instructor will not hesitate to go there. But since most of us are going to be quite skeptical that a student would have no personal goals, motivations, or values in skiing, you can expect more inquiring to try to unearth those hidden motivations.
What do you want, CharlieP? I've had the pleasure of skiing with you and getting to know you a bit, so I have some general ideas about your skiing preferences and ideals. If I were to ski with you tomorrow, though, I'd still want to know what's new. What have you been working on, and why? What revelations and breakthroughs have you encountered since I last skied with you? What questions have come up? What's working for you? What isn't?
Addressing the needs of the student is the primary objective, and the single measure of success, of any instructor in a lesson. It is said (Kim Peterson) that "an instructor who meets the needs of the student cannot fail, and an instructor who fails to meet the needs of a student cannot succeed." And the first, critical step in meeting someone's needs is, of course, identifying those needs.
When you get down to it, these activities--identify and address needs--are the two, and only two, critical functions of an instructor. They even represent the correct answer to a question that has long appeared on our certification written test: "What are the two important activities of an instructor in a lesson?" (And you thought the answer was "talk about skiing" and "talk about ourselves," didn't you?)
And when it comes to identifying needs, there are some that the instructor can figure out by observing, and others that the instructor can glean only by asking you questions. Yes, we can see your movements and technique, and good instructors bring a great deal of training and experience to the art of "movement analysis." That's some of the expertise you're paying for. But unless you tell us, we can only guess at what questions and goals you may have, and what motivates you as a skier--and elsewhere. The more we know about you, the better we can help you progress. And no one knows you better than ... you! In sharing honestly what you know about yourself, you become a partner with your instructor in identifying your needs and achieving your goals.
Hence the questions. Indeed, helping instructors to ask better, more revealing questions is a big part of all effective instructor training programs. "What do you want?" or "what are your goals?" may or may not evoke the answers we need. "What do you like about skiing?" may do it for some people. "What other things--sports, activities, recreation--do you enjoy, and why"--can reveal deep and important truths about you. "What's the best lesson you've ever had?" And so on. One of my favorite questions goes something like this: "If you were to look back at the end of today and think, 'wow--that was the best day and the best lesson I've ever had'--what would we have done--and what would I have done as the instructor to help make it such a success?" Answer that one and I'll learn a lot about what your definition of "fun" and "success" are and what you value in skiing, and I might even gain some insight into how you like to learn, how to pace your day, what teaching styles are most effective for you, where and when we should stop for lunch, and much more.
Answer well. It's in your best interest!