It would seem to me that the act of choosing isn't but deciding which alternative is preferred. And so impossible to separate completely from prior conditions or influences. So the question would then become what does "determined" mean in your definition?

The definitions of 'determined' and 'influence' are at odds with one another in your definition.

If a person's choice is mostly determined by the state of their brain at the moment of choosing, would that count as a "prior condition"?

It would mean the choice is mostly determined by their "self", but the definition you give isn't clear about whether that would count as free will or not.

so then what else informs a free agent's choice?

And if you have a free agent without prior conditions or influences - how was it made?