Defining engagement for business
Neuroscience might define engagement as the willing or unwilling allocation of neural resources along specific sensory channels. The two most obvious sensory channels are vision and hearing.
The individual who shuts their eyes to better hear some music is "engaged" by the music. Although this may not seem like predatory behavior, the same neural reward pathways are active as with the cat that is about to pounce.
Likewise, the guy who turns off his car radio so he can locate a specific address or street is "shutting" his ears in order to better see. He is "engaged" in locating an address.
This allocation of resources -- engagement -- is where profits happen.
Engagement resources are located in the cognitive, behavioral/effective, and motivational ({C,B/e,M}) brain regions. We know engagement is occurring because specific and easily measurable cognitive, behavioral/effective, and motivational neural activity is taking place.
These neural activities overlap, and these activities take place in the brains of everybody on the planet. When I'm on the court, I have the same {C,B/e,M} matrix as a world class tennis player. The difference is one of degree: The cognitive effort of the world class player is much higher than mine, and her motivation is greatly different -- a world title versus a Scotch in the club house at the end of the game.
The muscles and motions involved -- behavioral/effective element -- are also the same, except the world class player's is highly tuned due to their higher cognitive and motivational factors.
This is predation again. Whether a Scotch or a championship title, our brains are working towards a reward, and our {C,B/e,M}s are hunting down that reward.
Measuring engagement
Emotional engagement is the most easily measurable form of engagement. Emotional engagement is common to all {C,B/e,M}s and brings us back to being engaged to someone -- that form of predation that allowed our species to survive the 7 million years it took for that wiring to get in place.
We humans may think well of our rational minds, but any neuromarketer will tell you that emotions rule the world of commerce. The emotional mind's power has been demonstrated in everything from the Kennedy-Nixon debates to Baylor College's Pepsi-Coke challenge to Ogilvy's Rory Sutherland demonstrating that electronic cigarettes satisfy while nicotine patches and gum do not -- even though electronic cigarettes deliver no nicotine kick. Emotions may come into play at the end of the sales funnel (as with Audi purchasers) or at the beginning (Nissan Leaf purchasers).
Without emotional commitment, all you get is buyer regret and remorse at best and troublesome customers at worst.
The emotional mind's {C,B/e,M} control over our rational mind is best demonstrated by those non-conscious filters that are active when we're highly engaged. Abrupt interruptions usually trigger irritation that is demonstrated by an emotional response (we're rarely polite when we tell someone to "Shhh!").
High levels of engagement in meaningful noisy environments are called "high perceptual-load conditions."
Marketers rarely want to subject consumers to such things -- we don't want consumers working to pay attention. They'll fatigue rapidly (!) and go somewhere else.
So the goal of marketing is to create high engagement in "low perceptual-load conditions."
Marketing recognizes low perceptual-load conditions as someone using a browser, watching TV, and reading a newspaper or magazine. High perceptual-load conditions are people using their mobiles, tablets, and notebooks while commuting to work, at the cafe, at the beach, or in the bar.
There are lots of creative, usability, or just plain common sense reasons to create different content for mobiles, tablets, etc., versus browsers, print, and video.
But, before you get there, engagement reason numero uno is that the {C,B/e,M} requirements are so different from one medium to the next. What causes engagement in TV will be ignored on a tablet, and vice versa.