Hate to pull this old chestnut out, again, but I must…. it depends.
There is no one definitive definition of "lead UX designer." In some companies he or she may "own" the entire design of a product or a domain, and in other organizations the Lead UX Designer is simply the only designer assigned to a project. For example, I could be at Company A and have the title "Lead UX Designer, Mobile" and have the same role and responsibilities of a UX Architect or Director of Design. Or I could be the UX Lead ay Company B and I'm just doing detailed design documentation and wireframes… and not much else.
UX is a mature discipline, but there's still variety in the jobs and the roles… and I hope it stays that way, because (in my opinion) rigid titles and descriptions takes away from our ability to be agile and flexible about what we do. I'm all for standards, but not one that stifles design and innovation.
The one thing I think is constant no matter where you work is this: if you are the Lead UX Designer, you should… well, lead. Set direction, veto bad ideas, fight for quality. Negotiate for and try to produce the best experience possible. Fight for the user.