A good warm-up doesn’t have to make your body warm.
The authors believe that doing low intensity warm-ups such as treadmill running will not prepare your lower body for lifts such as heavy squats. Warm-ups should be specific to the muscles that are going to be worked and the manner in which they are going to be worked.
I do not agree with this new rule. I always begin my workouts with an easy jig jog on the treadmill. It gets my heart rate up, raises my internal body temperature by a degree, and gets my blood flowing. It suppresses my digestion (since I always eat something before lifting) and makes my body ready to do work.
However, as the authors state, there should also be a warm-up specific to the exercise that you are doing, and I agree with that 100%. Before I do squats or deadlifts, I will usually do a set of 15 with no weight at all, not even a bar. I just go through the motions and pretend I am holding the bar. I may look silly, but I feel that the dual warm-up better prepares me for the workout.
Normally, though, I only do a second warm-up for the first exercise I am going to do, which is usually some form of squat, deadlift, or lunge. Once in a while I will throw an empty set of something into the middle of a workout, but not very often.