I was a UX designer once, but looking back I was more like a product designer.
Now I am a product designer and I can tell that one title is co-dependent of the other, for me if you’re a product designer, you’re also a UX designer.
Today I call myself a service designer, a title doesn’t define, it’s your knowledge and how you apply that knowledge that defines you. Some people may argue, saying A UX designer will identify the problems, and a product designer will propose solutions. Oh well, then you’re saying a UX designer is focused in the problem and the product designer is focused in the solution? A pessimist and an optimist tale. I like that.