It doesn't mean that you let your Id rule and disrespect everyone around you. It means to do things natural to you and in line with your instincts.
Have you ever seen a really effective kindergarten teacher? Or salesman? Or engineer? They are naturally inclined to do what they do and have the instincts through education and experience to just do what others need to think about.
The example of the general manager at HP was a great one. Being a genuine and sincere person, even with subordinates is fine in a small setting. But when you're addressing 5000 employees, its formal by necessity and you're not speaking as you -- you're an officer of the company.
I think it's great when people don't try to be someone else. One of the people who I mentored for awhile started out as a real jerk... he almost got fired for attitude. It turned out that he was in the wrong role, an operational, follow the process role that was stifling to him and driving people crazy. We recognized that, got him in a better gig and he shined. If he had sucked it up and marched like a good soldier, I never would have known.