This is how I feel about "face recognition" (it should mean recognizing whether something is a face or not), but it is common to use eye tracking this way.
Where are you getting this language point from? If I look up any company selling "eye trackers", their products are all meant to track where you're looking, e.g., https://www.tobii.com/
Interesting. I picked it up working with Tobii devices a few years ago actually. I guess they updated their vocabulary.
Well then it's both?
Could be, but not necessarily. Eye tracking usually means it's tracking the eyes of one or more people in a video. Gaze tracking usually requires your eyes stay pretty steady and close to the tracker.