I'd say that a false approach, and that it doesn't capture the actual thing we mean by "free will" even if it seems it does.

I obviously have to disagree and I want to stress again that I am talking about the way the term free will is commonly used, not about the way it should be used. Go and ask people. I argue that most people will say that they could have made either choice in the sense that repeated trials would yield differing choices and that making the choice is not merely an act of flipping a coin.

This of course turns out to be an impossibility under close inspection, as I argued before, and it therefore follows that free will in the common usage of the term does not exist. But I am pretty sure most people will nonetheless object if you argue that they could not possibly have made any other choice than the one they made besides maybe by flipping a coin.

Neither dependent on the state of the universe nor random basically implies a soul. That is, an agent outside of the universe (that can still know part of the universe's history as it pertains to them and consider it when they take their decision but is not influenced by it directly to the decision). In the end, that could just be a random choice.

Such a kind of soul would be part of the universe as far as I am concerned. It is something that can be acted upon by the normal universe to record facts and it can act on the normal universe by influencing decisions. I am unable to see how such a kind of soul would be significantly different from a halo of weakly interacting dark matter surrounding every body, gravity aside.

There are weak bidirectional interaction between the normal matter of your body and the dark matter of your halo, and they are so weak that they escape our current detection capabilities. We also do not currently understand and are unable to observe the internal dynamics of dark matter halos.

Is there any aspect to souls that this analogy does not capture? Well, souls could possibly exist outside of space and maybe even time, but that seems to be nothing more than a minor technical detail, at least in the context of free will. So I argue that souls of the kind you described are not adding anything to the picture, they are mostly equivalent to some kind of yet undiscovered physics.

The essential when we say free will is that it's "me" the agent that decides.

It is undoubtedly true, that it is an important part of free will, that the decision originates from the agent. This aspect is, I argue, captured in the term will, the desires of an agent.

Some might think that this means we can decide arbitrarily (e.g. if we could repeat the experiment with all other conditions absolutely the same, we could possibly see a different decision), but I think that's a wrong extension of the basic requirement: that the decision is uniquely on the agent.

This is, I argue, the question of what the term free means. As argued before, I think most people actually understand it in the sense that you consider a wrong extension. I also think that in your point of view free would be mostly redundant, just affirming what will essentially already conveys, that the decision originates from the agent, that the decision is free from external influences.

And the way that this is ensured (and in line with both determinism and randomness in the universe) is that the agent is the decision (the sum of their decisions) embodied. Which is also kind of obvious: given the same circumstances exactly (to the last particle being the same etc) we would make the same decision. But it's also our history of passing through those exact circumstances and taking those decisions we took that makes us "I".

Here I just want to note that this interpretation might have its very own problems, especially what constitutes external influence? What if steak is out? Or if the restaurant is on fire? What if I force you at gunpoint to order fish? What if I kidnapped you twenty years ago and only gave you fish for four years so that you hate fish today? What if you are under the influence of drugs?

It might turn out to be quite tricky to find the boundary between agent and external influences in space and time and I would not be to surprised if, upon close inspection and careful definition, it turns out that the agent is identical with the past light cone of the decision.