IMHO, a "good" horse/human relationship is one in which the horse recognizes the human as the leader of the herd of two. That doesn't mean the the horse is always perfectly obedient or behaves like a robot. Normal horses will always "test" the leadership ability of higher-rung horses and people, to make sure that they are up to the task. The frequency and intensity of that testing depends on the individual horse and human.
That relationship is built on trust that is built on respect. It's not all sunshine and butterflies, but, for the horse, the ideal leader is one that is fair and consistent in expectations and requests, thus eliminating confusion. Boundaries are clear, the right thing is easy and the wrong thing is hard. It's similar to a healthy parent-child relationship, IMHO.
As far as little things like whinnying at "their" human's approach and the like, that depends on the individual horse and his idiosyncrasies. A horse/human team can have a wonderful relationship, and the horse might be aloof in behavior. I've seen some owners with very buddy-buddy horses that were horribly disrespectful and did not have what I would call a "good relationship."
I would characterize my relationship with my horse as as good, but not extraordinarily "special". Although he is still what I would consider green, he is very respectful and well mannered in general, and very rarely tests my leadership. As far as whether or not he enjoys my presence, he does whinny when he sees me, and seems to genuinely enjoy his job. Would he cross a mile of broken glass to get to me? Probably not. Does he "love" me? Hard to say, as who really truly knows the type and extent of equine emotions. He honestly tries everything I ask, and that is plenty for me.