They both are correct, although the one with Coleman where he moves slightly doesn't really add much, and does add slightly higher risk.
The theory is that by leaning forward on the eccentric phase you can get a greater stretch on the lats, and thus a better exercise. However, what you are really doing is simply changing the angle of the upper arm in relation to the upper body, and getting slightly more activation out of the lower back muscle groups.
If you watch carefully, the arms don't really start pulling in the Coleman video until his back is almost straight again, and they end in the same position, with upright back, shoulders pulled back tight, and elbows close so the grip is snugged in against the body.
The main differences is that by allowing the flexion forward, you can get an increase in the momentum of the weight, as it is already moving by the time your arms get into the act. If you are going for high weight, this can help, but it does increase the risk of injury because of increased activity/strain on the lower back. Assisted higher weight exercises and strict form exercises both have their place/uses, definitely what you want to avoid is going past vertical away from the weight stack in either instance.