Free will, could-have-done otherwise, and causal determinism

One common notion of free will is the idea of choice: you have a number of options in front of you, and you pick one. Or, perhaps a better way to talk about this is like so: you have free will if you could have done otherwise than you did. A provisional definition could be this:

(T1)  For any person x who performs an action A,
x performs A freely = iff
x could have done other than A. 

But that's not really good enough, because a determinist who believes that our actions are causally determined by our prior histories could agree with it. Such a determinist might say: 'of course you could have done otherwise, if you had a different history.'

Well, sure, I could have done otherwise if I had a different history. But that's still deterministic: one history determines me to do certain things, and another history determines me to do other things. And that's not really what I mean when I say that I could have done otherwise. What I really mean is that I could have done otherwise, even with the same history. So we could revise T1 something like this:

(T1*)  For any person x who performs an action A,
x performs A freely = iff
x could have done other than A,
given the same history.

That seems a little better.