That's some careful analysis!
Two remarks:
"Can" is the opposite of "unable". "Unable" means that the change involves granting ability to they who would act, i.e. teaching a technique, providing a tool, fixing the body, or altering the environment.
That's a good characterization, though arguably not a definition, as it relies on "ability", which is circular. I can do something = I have the ability to do something. I can = I'm able to.
But we can use the initial principle (it really needs a name) which doesn't mention ability:
You do a thing iff you can do it and you want to do it.
"Iff" behaves similar to an equation, so we can solve for "can", similar to algebra in arithmetic. I don't know the exact algebra of "iff", but solving for "can" arguably yields:
"I can do X" iff "If I wanted to do X, I would do X"
Which uses wanting and a counterfactual to define "can". We could also define:
"I want to do X" iff "If I could to do X, I would do X"
Though those two definitions together are circular. Maybe it is better to regard one concept as more basic than the other, and only define the less basic one in terms of the more basic one. It seems to me that "want" is more basic than "can", so I would only define "can" in terms of "want", and leave the definition for "want" open (for now).
Regarding the Confusing Cases. There are at least two canonical classes: akrasia (weakness of will) and addiction. "Can" the addict quit smoking? "Can" the person suffering from akrasia just do The Thing?
A better question perhaps: In which sense is the answer to the above "yes" and in which sense is it "no"?
One possibility is to analyze these cases in terms of first-order vs second-order desires. The first-order desire would be smoking or being lazy, the second order desire would be not to have the (first-order) desire to smoke, or be lazy. Second-order desires seem to be more important or rational than the first order desires. The second-order and first-order desire are "inconsistent" here. If the first-order desire to smoke is stronger than the second-order desire not to have the first-order desire, I don't quit smoking. (Or don't overcome my laziness in case of akrasia). Otherwise I do.
Here, the "can" definition as "If I wanted to do X, I would do X" is ambiguous. If it means "If I had the first-order desire to quit smoking, I would quit smoking", then the sentence is true, and I "can" overcome the addiction (or the akrasia). If it means "If I had a second-order desire to not have the first order desire to smoke, I would quit smoking", then the sentence is false, as I do in fact have the second-order desire but still don't quit. So in this sense it's not true that I "can" quit.
A similar but different different analysis wouldn't phrase it in terms of first and second-order desires, but in terms of rational wishes and a-rational urges. I wish to quit smoking, but I have the urge to smoke. I wish to get to work, but I have the urge to be lazy. If we count, in the definition of "can", urges as "wanting", I can stop smoking, if we count "wishes" as "wanting", I can't. By a similar argument as above.
I think these cases are actually not a major problem of the "can" definition. After all, it seems in fact ambiguous to ask whether someone "can" overcome some case of akrasia or addiction. The definition captures that ambiguity.
Followup to: Will issues are quite nearly skill issues
You do a thing iff you can do it and you want to do it.
Something you can do, but didn't, looks just the same to the simple observer as something you can't do. Likewise with what you want to do, but didn't. When you don't do something, obvious physical reality leaves ambiguous whether you were unable or disinclined.
Here are some things I don't do:
Each of those things, I don't do for a single reason: I'm unable to do the first three, and disinclined to do the last three. To categorise them like that is intuitive. The intuition here, made precise, relies on counterfactuals. "Unable" and "disinclined" each classify answers to "what would have to change to make you do that?".
"Can" is the opposite of "unable". "Unable" means that the change involves granting ability to they who would act, i.e. teaching a technique, providing a tool, fixing the body, or altering the environment.
If you told me where the airport is, or gave me a jetpack I trust as safe enough, I would fly, at least occasionally. If you modified my brain to consolidate learning and my body to recover while awake, and pointed me to a large supply of stimulants, I would gladly stay awake for a full year, maybe more. If I studied malicious hacking methods, and you told me the network address to access Instagram's internal database and server controls, I would wipe their data and shut them down.
I can sneak into the gym, or use a chalkboard, or crush my finger, insofar as teaching and equipping me wouldn't change that I don't do any of those.
"Want" is the opposite of "disinclined". "Disinclined" means that the change involves convincing or incentivising the one who would act.
If those at the front desk of the gym turned off the card-scanner and ignored visitors walking in, my seeing that would convince me to abandon inhibition and run past. If you invite me to lecture in the room where I already have chalk and board, and I see value in doing so, I'd write on the chalkboard. If you credibly offer a suitably large amount of money for me to crush my finger, I'd go slam a door onto it.
I want to fly, stay awake for years, and wreck Instagram, insofar as talking me into or bribing me for any of those would end with me still not doing those.
If "can" and "want" are orthogonal, there should be some actions for each combo:
The first combo entails those things you actually do. The second and third, I exemplified above. An action in that final category would take multiple changes, some ability-granting, some convincing. For example, in my case:
To get me to play guitar, I'd need to find a guitar, and learn how to play. But I could only learn if I had strong interest or incentive to play guitar, the provision of which would take another change.
Cookies no longer appeal to me, so I would only bake them to give to others. That only happens if someone offers to buy them, or I am convinced to prepare a gift, which I don't do habitually. Besides the matter of interest, I'd need to learn a recipe, as I've little baking experience, and get ingredients.
I like my friends, and so prefer that they stay alive, more than the weaker preference for life I apply to people by default. To override that would take circumstances which change me from liking a friend to hating them. That only leads to murder if the hatred, plus fear or outside incentive, ends up deep and great enough to outweigh my moral qualms and fear of punishment. But, additionally, I'd have to find weapons and/or tactics which are deadly enough.
"Know how to" is a subset of "can". We can define it by its negative, "don't know how to", which is a special case of "can't": you "don't know how" iff an act of learning is part of the minimum counterfactual change to make you act. So I know how to hard-boil eggs, yet usually can't: the change from this world to the one where I hard-boil eggs is to give me raw eggs, a pot, and access to a stove, with no learning involved.
Likewise, "allowed to" is roughly a subset of "want to". Its defining negation, "prohibited from", is roughly a special case of "disinclined to": you're "prohibited from" an action iff the counterfactual includes a change to the policies others maintain. I say "roughly" sith people sometimes do things which are prohibited. Policies affect actions via negative incentives and qualms, a subset of those changes that take people from "don't want" to "want". I'm allowed to stay awake for a 30-hour stretch, but don't want to: arguing to me that it's a wise move may make me do it, as would offering a reward for it, but no jurisdiction I'm in has policies which, if removed, would lead me to stay awake much longer.
I don't want to slit my own jugular vein, nor do I know how to. A counterfactual could correct for both of those: I'd want to if I had a large incentive for it, and assured medical intervention, such that I'd survive with sufficiently mild complications. I'd know how to if I studied the relevant anatomy.
When the cut itself is all that remains, my survival instinct would cause quite the hesitation. It might completely inhibit me from such an oft-deadly cut. Survival instincts happen from the brain, and so may seem a matter of "want". I'd say they're a matter of "can" — I can't cut my jugular — biting the bullet of the definitions here, I anticipate that an increase in the incentives would leave the result the same; the changes under "don't want" don't suffice.
On several occasions, people have left phones or laptops near me. Each time I was alone, I knew how to pick up the precious device, and take it home or hide it, to keep it for my profit. Yet I never do that, in part sith theft is illegal and often punished.
Were theft made legal, I'd still have morals, intuitive or taught, which oppose theft. Morals, more than my other preferences, I might try to enforce on other people. If self-enforced morals are a "policy", then, with externally-legal theft, I still would be "not allowed" to steal. Does a rule you apply most strongly to yourself count as a policy?
"Liking" an action arises from your views, all mental, towards the action. "Wanting" has the same basis. The two differ in whether they directly affect which actions you take: only wanting does, and so only wanting is a category of possibility. Liking just correlates with possibility insofar as we often learn to want those things we like.
Probability is also separate from these categories of possibility.
Intuitively, I can take down Instagram more than I can fly more than I can stay awake for months. I want to write with chalk more than I want to break gym entrance policy more than I want to crush my finger. The magnitude of a partial "can" or "want to" is inversely proportional to the scale of the salient counterfactual change that would cause the action. Telling me a few IP addresses and passwords is a smaller change than lending me a plane is a smaller change than safely and permanently suppressing my urge to sleep. Having me lecture is a smaller change than making gym staff abandon their tasks is a smaller change than incentivising me enough to crush my finger.
The magnitude of a probability is a dot product between the probabilities of future counterfactual changes and the probabilities that they would each lead to the event being studied. Possibility — "can" and "want to" — are defined from the most salient routes; probability is defined from the most probable routes. So e.g. crushing my finger is the most probable of the three I don't want, for a bad lapse in my attention would also suffice, much more commonly than people paying me to injure myself.
I have broken down intuition into different intuition, perhaps more easily scrutinised: why are the counterfactuals I listed the most salient ones?