In art programs, you can 'manipulate' selections of images in various ways.

Some examples of manipulation operations are 'Skewing', 'Shearing', 'Scaling', and (possibly) 'Rotating'.

I understand there might not be a "standardized" term, but I was wondering if there is a commonly-used term for highlighted manipulation/operation?

What's happening in the image is the user grabbing a single corner of the image and 'stretching' (!) that one corner independently of the other three corners.

Paint Shop Pro just calls it "Free" in a drop-down menu {Scale, Shear, Perspective, Free}. I think I'll just call it 'Stretch' unless someone has a better name.

If I was to rename 'perspective', I'd probably call it 'Tilt' - might be too late for that.

I like names to all be verbs that have past tense versions as well: Scale -> Scaled, Shear -> Sheared, Tilt - > Tilted, Stretch -> Stretched.

If it's only about moving a single vertex, I'd simply call it a "freeform" manipulation. That is, if the fact that the texture in your screenshot is also stretched is merely a byproduct of the vertex being pulled out.

If it is really about dragging a vertex belonging to some kind of control cage and modifying the "inside object", that would be a "lattice deformation" or a "lattice modifier", like this:

For Photoshop, that would probably be included under perspective warp, though distort is also a related term.

warp

http://helpx.adobe.com/photoshop/using/transforming-objects.html (2:35 of the video)

http://helpx.adobe.com/photoshop/using/warp-images-shapes-paths.html

If it's only about moving a single vertex, I'd simply call it a "freeform" manipulation. That is, if the fact that the texture in your screenshot is also stretched is merely a byproduct of the vertex being pulled out.