Both of your images show examples of a grimace:
A contorted facial expression, often expressing contempt or pain.
The noun is to grimace:
To make grimaces; to distort one's face; to make faces.
Grimacing indicates pain or distress, whether one's own or in sympathy with another's. It is usually made by stretching the entire side of one's mouth. If instead you want to indicate disgust you can use the phrase curl one's lip [in disgust], as you discovered. This action generally uses only part of one lip, rather than the whole mouth.
Sources: Kaitlyn on Pinterest, Micro Expressions Training Videos
This is very similar to a sneer, with the difference that "sneer" has the connotation of contempt or scorn while "curl one's lip" has the connotation of disgust.