While I don’t agree with your final conclusion (that no one should call themselves UX Designer), I think your reasoning is on point. We can agree or disagree on the title as professionals, but it also depends on the industry, including people that don’t necessarily think about the title itself. Recruiter or a person from HR will probably stick to the term that is most widespread and your chances to get considered for a job in the UX field are much better if you use the accepted term (even if it is not a correct one). I think of it like a nickname — you can choose one for yourself, but you will always be called by a nickname that is widely accepted.
On another note, the title of Product Designer has a problem, too. Apart from being too broad, because a product is not necessarily digital, and in most cases a single person is not responsible for creating a product, the same way one person is not responsible for user experience.