Would a corner lip lift help my downturned smile? (Photo)

I don't actually want my natural lip shape lifted or altered -- only excavated so it can be seen LOL. On the right side of my face, some tissue has descended (I'm in my 40s) and is literally burying my lip corner. My skin is in great shape and I'm worried that if a very sizeable triangle of skin were removed, it would leave bad scarring. My questions are -- is it considered a lip lift even if skin is just being removed? Would this procedure be with a scar in my case?