Why does the lip lift not lift the sides of the mouth?

Many thanks for posting this question. As the scar from an upper lip lift is confined to the base of the nose, the extent of the lift on the upper lip is in part determined by the width of an individual's nose, and in part by the width of their mouth - someone with a broad nose and a narrow mouth will have a greater width of their lip lifted, and vice versa.Therefore the width of lip lifted can vary between individuals

The traditional bullhorn lip lift removes the largest amount of tissue from the central area creating a central lift that tapers at the ends.  There is an alternative called an extended bullhorn lift, which extends laterally.  When I perform this technique I take more laterally than centrally.  For example, I will take 10 millimeters laterally and only 4 millimeters centrally.  This is done to pull up the corners, which are frequently depressed.  In older women this is essential, otherwise you will end up with a bunny deformity, which looks cute on young women and inappropriate on older women.  The extended lip lift does 4 things: shortens the distance between your nose and upper lip, increases the mucosa of the upper lip, increases incisor show and pulls up the corners.  Best Wishes,Gary Horndeski, M.D.

There are simply various methods of lip lift, among others the following; 
Bullhorn lip lift – focuses on the central part of the lip.  It reduces the distance between upper lip and nose base in the central area. It does not apply to corners.Corner lip lift – surgeon lifts lip corners. Gullwing lip lift – based on removing a strip of skin from the area above the lip. 
best regards, 
Piotr Osuch, MD, PhD

By the location of the skin excision pattern, it can not lift any upper lip  vermilion edge that lies outside of a vertical line dropped down from its most lateral excision end. Even the so called extended subnasal lip lift can not do it. The only way the lateral lips gets elevated with a lip lift if it is combined with vermilion advancements on each side of the lip. You are astutely correct in that it should be known as a central or Cupid's bow lip lift.