What UX designers do: Products should create a smooth experience for customers. Not a bumpy road, not a mine field. An awesome user experience designer considers all touch points to optimize product engagement and delight--something I call "flow".
Scope: Proper UX involves visibility across departments and ties into the development process, back-end architecture, customer service, & advertising experience. This is the one person in your company who keeps her/his eyes focused squarely on the enduser.
In other words, pretty images & icons are nice. But being able to checkout quickly is totally awesome. Get it?
Success Measures: Once users exit the flow, user experience designers want users to remain enthusiastic, recommending, & sharing the spoils of the product, with others.
I say nothing about the tools, deliverables, or methods but I hope I provide a new spin on something I've, in the past, struggled to describe.