UX is not about polish. UX is the product -- literally, from the user's perspective. I don't think there is a meaningful difference between Product Engineer and UX Engineer.

The X in UX is Experience so indeed a UX Engineer's role would be to improve the user's experience of the product not necessarily define what the product should be. It's unfortunate that UX is being confused with Product. Let me illustrate with an example: Consider a product with 20 features, 10 of which will be helpful to the user once they have been using the product for a while. The UX engineer's job is to identify this and provide a user flow of progressive disclosure of those 10 features based on user's actions. This improves the Experience for a user by not overwhelming them with all 20 features when they are just starting to use the product.

A Product Engineer's job is to identify the 20 features because they are useful for users.

In terms of conventional job titles, that's the job of the Product Manager.

True. And I mentioned in a comment above that a technical Product Manager would be a Product Engineer. However a non-technical Product Manager would not be.

But how much technical skills do most UX Engineers have? Seems from the general consensus Production Engineers touch software more.