i.e. Holding everything constant (including X's mental states), X could nevertheless have done differently. Even if his beliefs, desires, and reasoning were exactly the same, a different decision may have resulted.
i.e. X could have done differently if he had wanted to.
Let's examine these alternatives by way of an example:
Suppose I'm driving along the motorway, when the thought crosses my mind that I could swerve across the road and crash into the oncoming traffic. I have no reason to do such a thing (for I believe that doing so would cause injury or death, and I have a very strong desire to avoid injury or death). But suppose that, after going through the reasoning which surely should result in my decision not to swerve... suppose instead that I (inexplicably) decide to swerve... I die moments later.
Does that sound like a free action to you? I really don't think it is. I would actually feel much more free if I knew for sure that my actions were reliably caused by my beliefs and desires. I would feel much more free if, in that given scenario (i.e. given my current beliefs & desires) , it were truly impossible for me to kill myself (and possibly others) like that. The possibility that I could go through all the reasoning necessary to reach a good decision, and yet have the opposite decision spontaneously result... it sounds more like some kind of mental malfunction, than 'freedom'.
I think that sort of example demonstrates quite nicely the advantage of F2 over F1. I am free (according to F2) because I could have swerved if I had wanted to. But I didn't want to! Given the fact that I didn't want to, surely I can only be said to be free if my actions conform to my desires in a rational manner. Surely I am only free to the degree that my desires deterministically influence my behaviour. That is to say, surely I am truly free only if (given my beliefs & desires) it is ensured that I would not have swerved at that moment.
Freedom requires that our actions be caused by our mental states (beliefs, desires, etc). Yet F1 seems to require that our actions be uncaused (or, rather, caused in some unpredictable, or "random" manner). The indeterminate causation required by F1 thus strikes me as being not freedom at all, but rather, an obstacle to freedom's realisation.
I think what the libertarian needs to do (if he wants to hold onto a coherent notion of freedom) is modify F1 to allow actions to result determinately from our mental states, whilst simply insisting that our mental states not be externally determined.
To clarify: the compatibalist (F2) merely requires that our desires (etc) be the proximate cause of our actions - but it's okay that our desires were themselves caused by preceding events that were outside of our control. In contrast, the libertarian (F1) would require that our mind be the ultimate cause of our actions. The mind itself - according to this view - must be somehow uncaused ("transcendent", perhaps), and "free" from external influence.
Put like this, the libertarian view at least makes a bit more sense. But it asks the impossible. Our personalities don't come from nowhere... we are hugely influenced by both nature and nurture - our genes and upbringing - both of which are external to us, i.e. outside our control. If that makes us unfree, then freedom is an impossible ideal, which has never been attained.
Alternatively, we could choose F2 - a concept of freedom which is fully consistent with determinism, yet nevertheless provides a useful and plausible account of free action (i.e. action as a result of our mental states and reasoning).
Update: see also Jason's excellent post, Evil Robots, which shows how a machine could have free will.