Well said Alexis, I agree with the end of your post and that was actually what I was aiming at, but I disagree that stretching should be done only after training. Maybe you were referring to static stretching, but there are other forms of stretching, such as dynamic stretching, which should be a part of a warm up.
A quote from the very same book you mentioned:
Stretching is not warming up! It is, however, a very important part of warming up.
Warming up is quite literally the process of "warming up" (i.e., raising your core body
temperature). A proper warm-up should raise your body temperature by one or two degrees
Celsius (1.4 to 2.8 degrees Fahrenheit) and is divided into three phases:
1. general warm-up
2. stretching
3. sport-specic activity
It is very important that you perform the general warm-up before you stretch. It is
not a good idea to attempt to stretch before your muscles are warm (something which the
general warm-up accomplishes).