We can linguistically analyse "free will" to explain what the most common interpretations of the term are, but we'll always miss something.
There are two completely different problems we can try to solve.
The first problem is, "what do people mean by the words 'free will', and what do they believe about the concept they reference, and why do they use these words or concepts to begin with"? This is a problem for linguists, anthropologists, historians, psychologists, neurologists, maybe even evolutionary biologists. And it's a completely empirical one, even if hard to solve.
Crucially, it doesn't seem to be a problem in need of philosophy, unless philosophers are somehow selected or self-selected so that their own self-reports are more valuable than those of other people, while still being representative of people in general. (Nobody really wants to know the answer to "'what do philosophers use the words 'free will' to mean, which is completely unrelated to what everyone else uses those words to mean?")
I hope everyone would agree that whatever people mean by words, and believe about their referent, doesn't necessarily prove any objective truths about the referent.
The second problem we can try to solve is whether there actually is free will and how it should be best described. Of course, if different people mean different things by "free will", then this is a set of different and unrelated problems. Since we're presumably motivated by actual problem(s) and not by the mere occurrence of these words - even if the problems are driven by intuition and not by empirical evidence - we should be able to taboo the words "free will" and state the problems directly. Solving the second kind of problem should not require first solving the first one.
Have philosophers done this? Did they turn out to be working on the same questions? Do they have any good arguments for any of their positions beyond intuitions (which should be used to solve the first kind of question, not the second one)?
ETA: for example, a dualist might say: "spirits can affect matter, creating material effects that don't have detectable material causes. This action is called the free will of the spirit, particularly a human spirit." This might be a strawman which no-one actually believes, but it would be a concrete statement about free will, trying to solve the second kind of problem.
Of course, everything I've said here about free will is applicable to any philosophical debate where not everyone agrees about the meaning of the words being debated.
In your first case, you gave a classic example of experimental philosophy. I completely disagree with your characterization of the subject as non-philosophical . Proto-philosophical , I'd suggest. Concept mapping is an excellent starting point for recommendations on how to improve our understanding of problems that involve substantial conceptual confusion.
Google x-phi (yes it has a nickname ) and you'll see plenty of work on free will, as well as plenty of metaphilosophical debate on its usefulness.
Up voted - great comment regardless of the above disagreement.
One of the great challenges of learning philosophy is trying to understand the difference between different schools of thought. Often it can be almost impossible to craft a definition that is specific enough to be understandable, whilst also being general enough to convey to breadth of that school of thought. I would suggest that this is a result of trying to define a school as taking a particular position in a debate, when they would be better defined as taking a particular approach to answering a question.
Take for example dualism and monism. Dualists believe that there exist two substances (typically the material substance and some kind of soul/consciousness), while monists believe that there only exists one. The question of whether this debate is defined precisely enough to actually be answerable immediately crops up. Few people would object to labelling the traditional Christian model of souls which went to an afterlife as being a Dualist model or a model of our universe with no conscious beings whatsoever as being monist. However, providing a good, general definition of what would count as two substances and what would count as one seems extraordinarily difficult. The question then arises of whether the dualism vs. monism debate is actually in a form that is answerable.
In contrast, if Dualism and Monism are thought of as approaches, then there can conceivably exist some situations Dualism is clearly better, some situations where Monism is clearly better and some situations where it is debatable. Rather than labelling the situation to be unanswerable, it would be better to call it possibly unanswerable.
Once it is accepted that dualism and monism are approaches, rather than positions the debate becomes much clearer. We can define these approaches as follows: Monism argues for describing reality as containing a single substance, while dualism argues for describing reality as containing two substances: typically one being physical and the other being mental or spiritual. I originally wrote this sentence using the word ‘modelling’ instead of ‘describing’, but I changed it because I wanted to be neutral on the issue on whether we can talk about what actually exists or can only talk about models of reality. If it was meaningful to talk about whether one or two substances actually existed (as opposed to simply being useful models), then the monism and dualism approaches would collapse down to being positions. However, the assumption that they have a "real" existence, if that is actually a valid concept, should not be made at the outset, and hence we describe them as approaches.
Can we still have our dualism vs. monism debate? Sure, kind of. We begin by using philosophy to establish the facts. In some cases, only one description may match the situation, but in other cases, it may be ambiguous. If this occurs, we could allow a debate to occur over which is the better description . This seems like a positional debate, but simply understanding that it is a descriptional debate changes how the debate plays out. Some people would argue that this question isn’t a job for philosophers, but for linguists, and I acknowledge that's there's a lot of validity to this point of view. Secondly, these approaches could be crystalised into actual positions. This would involve creating criteria for one side to win and the other to lose. Many philosophers who belong to monism, for example, would dislike the "crystalised" monism for not representing their name, so it might be wise to give these crystilised positions their own name.
We also consider free will. Instead of understanding the free will school of philosophy to hold the position that F0 exists where F0 is what is really meant by free will, it is better to understand it as an general approach that argues that there is some aspect of reality accurately described by the phrase “free will”. Some people will find this definition unsatisfactory and almost tauntological, but no more precise statement can be made if we want to capture the actual breadth of thought. If you want to know what this person actually believes, then you’ll have to ask them to define what they are using free will to mean.
This discussion also leads us a better way to teach people about these terms. The first part is to explain how the particular approach tries to describe reality. The second is to explain why particular situations or thought experiments seems to make more sense with this description.
While I have maintained that philosophical schools should be understood as approaches, rather than positions, I admit the possibility than in a few cases philosophers might have actually managed to come to consensus and make the opposing schools of thought positions rather than approaches. This analysis would not apply to them. However, if these cases do in fact exist, the appear to be far and few between.
Note: I'm not completely happy with the monism, dualism example, I'll probably replace it later when I come across a better example for demonstrating my point.