Could someone provide a quote or two showing that Sam disagrees with any of the above? Steel-manning only a little, I believe Harris' goal isn't to find the One True Definition of morality, but to get rid of some useless folk concepts in favor of a more useful concept for scientific investigation and political collaboration. He antecedently thinks improving everyone's mental health is a worthy goal, so he pins the word 'morality' to that goal to make morality-talk humanly useful. Quoting him (emphasis added):

[T]he fact that millions of people use the term “morality” as a synonym for religious dogmatism, racism, sexism, or other failures of insight and compassion should not oblige us to merely accept their terminology until the end of time. [...]

Everyone has an intuitive “physics,” but much of our intuitive physics is wrong (with respect to the goal of describing the behavior of matter). Only physicists have a deep understanding of the laws that govern the behavior of matter in our universe. I am arguing that everyone also has an intuitive “morality,” but much of our intuitive morality is clearly wrong (with respect to the goal of maximizing personal and collective well-being).

I think this view is more sophisticated than is usually recognized. Though it's definitely true he doesn't do a lot to make that clear, if so.