How to handle a title (PostDoc) I consider misleading?
In my experience, "postdoc" isn't even treated as a title, and what your contract says is your job title is kind of irrelevant, because you can write it up however you want on your CV.  My contract says I'm a "scientific researcher," but I have what is generally considered a postdoctoral position (because it's after the doctorate), and I alternate between both descriptions depending on the context.  Outside of academia, no one really knows what "postdoc" or "scientific researcher" mean anyway, so I call myself a "mathematician"-- a title which isn't written on any official documentation anywhere.  Furthermore, no one is going to address you by the title on your contract.  No one has ever referred to me as "Postdoc Moonlet" or "Scientific Researcher Moonlet."  The only title I use sometimes is "Dr." (which is ok, because I have the degree to back it up).

As long as you're not telling people to call you a doctor, I don't think there's anything to worry about.