Taking a step back...

The Oxford English Dictionary offers the following in its definition: an agreement; a formal promise, contract, or pledge; to be appointed; an obligation; attachment; involvement, business requiring attention. The verb engage means: to expose to risk; to commit; to attract and hold fast; to occupy or employ; to fight or battle. Just about all of these terms may be employed in describing student engagement.

When considering the topic of Student Engagement  Chris Garrett (2011) posed the question  " Do we really have a clear idea of what that means?"  It caught my attention as I've taught students ( and been a student with classmates) that look on the surface to be ideal learners.   They participate, they complete all the assignments (sometimes early), they do moderately well to extremely well on exams... and all of that means absolutely nothing when considering their "engagement".
It means they are focussed, organized, motivated to achieve an outcome, calculating and strategic (in a non-sociopathic way) but will tell you honestly; they couldn't care less about the topic being taught and have no interest in it whatsoever.  Are they engaged?   Can a learner be "technically" engaged but not "really" engaged?

Garrett goes on to suggest that teachers consider their own definitions of and expectations for student engagement. Accurately detecting genuine engagement can be challenging.

Taking a step backwards from "Student Engagement"  to considering what engagement means to us...