Bewildering Belinda. Seriously, what you mean by determinist must be something different from what other determinists mean by determinism:
Macmillan Dictionary: "the belief that everything is caused by another event or action and so you are not free to choose what you do" [Emphasis mine]
American Heritage Dictionary: "The philosophical doctrine that every state of affairs, including every human event, act, and decision, is the inevitable consequence of antecedent states of affairs."
There is nothing wrong with giving words your own personal meaning, so long as, if you expect others to know what you are talking about, you explain that private meaning. No other determinist that I know (I admit I do not travel and the same circles as most determinists) believes that things can be both chosen and determined. The whole point of determinism is that it excludes choice.
If a choice is possible it means it is undetermined, that there is an alternative to any specific action that can be chosen. [A computer program that uses a 'decision tree' to make a 'decision' is not a choice, it is totally determined. When any animal does anything, it could never have done anything else. It's owner's ignorance of what it is going to do does not constitute a choice.]
I know your view is different, and I'm honestly not arguing with it, because I do not know what it is. Now you know my view, perhaps you'll explain how something can be determined and still be a choice. I suspect it is what you mean by, "choice," that is unique.