As you might imagine, I have a totally different method for warming up my clients. If I need them to get on a treadmill at all it is because I am not prepared and I want them to kill about 5 minutes time while I prepare for their workout. Not that the "cardio" is needed at all.
Two things you want you warm-up to do is to arouse your nervous system and do some "dynamic" stretches. I use a complex that I got yet again from Bill Hartman. It is called "Rufus Complexes". Do only 5 reps of each exercise. Don't have new clients do all these movements. Here is my warmup:
Using an empty 45 pound bar...
hang cleans (half-way down the quad)
Hang cleans (to the knees)
Hang cleans (below the knees)
hang power snatchs
bent over row
RDL
front squat
shoulder press
front squat push/press complex
squat
goodmornings
front lunge
crossover lunge
side lunge
rev lunge
Thats about 75 reps. It will get your heart rate up, it will move you in an ever-widening range of motion on just about every body part, and it will arouse your CNS and prepare it for a load. As Alwyn Cosgrove says about warmups, you body only needs to to know that it will be this range of motion, and that its gonna be heavy!
I don't even do any more static stretches anymore as they are obsolete. In fact, many recent studies (which I will cite if you need it) indicate that static stretching prior to heavy lifting REDUCES the amount of force production. If you need to stretch a part, wait till after the workout. Before, do your stretch under a load and with movement.
Also, on the actual exercise you are going to be lifting, Don't do a bunch of reps for a warm up... Pre-exhaustion only pre-exhausts... If you want to know what your actual power output is, then do a light warm-up with low reps as well. What is your goal? If it is strength/endurance the you will want to train in the 6-8 rep range. Therefore, in your warm-up, only do the amount of reps you intend to do for the set (6-8 reps). You only need to let the body know what the ROM is going to be. Do one or two heavier sets to really prepare, but don't do anything that is going to push it to the limit on a light weight, that might exhaust your muscle. Now you are ready for business!
I do agree that some rowing motion needs to be done. The lower traps are the weakest muscle in most people's bodies. Rows do a good job of working these, as well as the lats, rear delts, and rhomboids.