Intensity is effort IMHO. I rarely get to the point that I actually puke, but there have been many times that I feel the need to. Like today is going to be with my Markus Ruhl Shock Routine for legs. FUCK SQUATS. But will be intense nonetheless. Intensity is getting up from the bench or squat rack and feeling the rush of pain and adrenaline rush thru your body enough to make you throw your hands over your head and scream like Triple H. Thats intensity. You dont scream to sound cool, but because it is force out of your lungs.
THAT WHICH DOES NOT KILL ME, CERTAINLY MUST MAKE ME STRONGER
To me, intensity tells me that i'm rising above status quo by going to failure or beyond. If i'm going to failure, meaning i'm fighting like hell to eke out that last rep and am uncertain to achieve it, or if i then cross over into forced reps, negatives, cheats, etc., THAT'S intensity, that's what's going to make the difference and shock the body into growth vs. same ol same ol. Intensity and going through the motions should be at opposite ends of the spectrum.
IMO, intensity is the mindset you need, the mindset you get when you push yourself as far and further than you though you could go. When youre done with the lift, youre lightheaded and nauseous. When you get home, your eyes are bloodshot. Intensity is balls to the wall, 100%.