A lateral brow lift will raise the outer third of the eyebrow and limits the elevation of the hairline.
I think your question is about dealing with lateral hooding of the eye caused by sagging of the brow laterally. This creates a sad and tired look. This is usually handled by some form of browlift in concert with upper blepharoplasty. I prefer what I call a Modified Browlift as it is minimally invasive and successfully handles the problem of lateral brow ptosis and is easily combined with upper blepharoplasty. Most people are quite presentable and back to work in 7-10days. Cost varies based on combinations of procedures and geographical location.
A temporal brow lift is a limited brow lift that elevates the lateral brow and helps reduce the excess skin lateral to the eye, thus reducing the "crow's feet." It is an option when a full brow lift is not needed. It can also be extended with a superior incision to elevate the brow all the way to the medial brow where you really do not want any elevation. As far as brow elevation, this is as good as a standard brow lift. Both can be done without endoscopic equipment. If you desire elimination of the muscle between the brows, this can be done through an upper eyelid incision.
A temporal lift is similar to a brow lift but limited in length. It mainly focuses on elevating the lateral brow a bit.
A temporal brow lift is one based on a lift of the lateral brow only along the temporal line. This is actually above the temple and the lift of the brow is upward and slightly back. The technique avoids the scar and numbness in the central scalp, and also keeps the lift on the lateral rather than the central brow to prevent the surprise or hollow look of a full brow procedure.
Temporal brow lift vs vertical brow lift?? Gravity works vertically and therefore 35 years ago we developed the Irregular Trichophytic Forehead lift. The pull is up and down and you will not have a noticeable scar if properly done. Temporal lifts pull the hairline back and often create an "evil" look.
There are also many ways to accomplish the temple lift from a skin only dissection from the hair line to a full endoscopic forehead procedure where the emphasis of the lift is in the temple area. Ultimately there is not substitute for a personal consultation with an oculoplastic surgeon who specialized in surgery of the eyelid, forehead, mid face, and face. The American Society for Ophthalmic Plastic and Reconstructive Surgery maintains a geographic directory on their website that can help you find the right surgeon (ASOPRS dot org).