Someone here with more knowledge than I should confirm or deny what I learned at a dojo decades ago, namely that the angle of bow is of significance. I was taught that exposing the nape of the neck (in essence offering the neck to a sword blow) was to affirm obeisance to the master, who had the power of life and death over you. I was taught to always keep my forehead up.
As I recall, Mr. Miyagi made the same point (in a physical way) to Daniel-San.
An inferior bows the lowest to his superior. Equals bow equally. Americans bow to no man.