What the Hell is "Free Will?"

I've been seeing a lot of talk about free will lately, and I have to confess that I honestly don't know what that term is supposed to mean.  It's as if we all have intuitive "gut sense" of what free will is supposed to be, but then it always breaks down whenever we try and give it a hard definition.  Now I honestly want to know what free will is, and how to recognize a free agent if and when I should ever encounter one, but for some reason, nobody ever seems to give me a straight definition.  So to help cut through the ambiguity, I'd like to offer a simple philosophical challenge to everyone out there who sincerely believes in free will.  It goes like this:

Imagine yourself in a room, sitting at a table.  Across from you are what appear to be two identical twins.  They look the same, they act the same, and in all physical respects, they are as alike as two people can possibly be.  However, there is one key distinction that exists between them: one of them has free will, while the other one does not.  For example, maybe one of them had a brain aneurism that robbed him of the free will center in his brain.  Or maybe one of them is a kind of artificial organism, created in a lab, and programmed to look and act like a person in every superficial way, but without the "free will" part - that sort of thing.  Your only job is to tell me which is which.  Which one has free will and which one does not?

If that sounds a little unfair, then let's make it easy.  I'll grant you access to any measurement equipment you like.  If you need a functional magnetic resonance imager, then fine.  You can have one.  If you need to dissect their brain, then go right ahead.  Or if you need a mass spectrometer, then that's great too.  Anything you like, just so long as it is at least possible to access in principle.  For instance, time machines are obviously not an option because they they simply don't exist, and almost certainly never will.  I also cannot accept an answer involving other immeasurable, supernatural forces, like "the one with the soul has free will."  Otherwise, we'd just have to repeat the entire experiment over again to figure out what a "soul" is.  So unless you're prepared to go down that road, then just stick to the question at hand - what set of empirical data would you need in order to objectively measure the difference between a thing that has free will and a thing that doesn't?

Now I've gotta tell you, I've posed this problem before, and it is like pulling teeth to get a straight answer out people, especially when it comes to Christians.  For example, one very common response I've encountered is the idea that free will is somehow "immaterial," and therefore cannot be empirically quantified.  And that's fine if you want to think that, but it's also an open admission that free will effectively has no meaning. You've basically just told me that there is literally no way to differentiate between things that have free will and things that don't.  I could claim that rocks have free will, or that Canadians don't, and there would be no way to prove either of those assertions false.

Then on the rare occasions when I do find people willing to offer a coherent definition, they almost always refuse to follow it consistently.  For example, most common definitions of free will tend to involve things like "the ability to choose between different possible courses of action without coercion," or maybe something similar; which is again fine, until you realize that computer programs are also perfectly capable of meeting that exact same criterion.  Or maybe if not today, then soon, because it's really only a matter of time before artificial intelligences begin to mimic the cognitive capacity of human beings.  Yet once you start going down that road, most people will immediately begin to back pedal, saying that robots are just collections of transistors and therefore cannot possibly have free will.  And that's fine if you want to think that, too, but all it does is bring us back to the original challenge.  How do I tell the difference between a thing with free will and a thing without?  Why do sophisticated networks of neurons in human brains get to have free will, but not sophisticated networks of silicon transistors?  What's the difference?  Because again, if you cannot tell me the empirical difference between a thing with free will and a thing without, then there simply is no distinction between those two states.  It's just a word that has no tangible referent.

Thank you for listening, and I look forward to hearing your answers.