I'm a newbie here and just wanted to add a comment/observation. I love this topic!!
Being a owner of a dog boarding facility I get to observe dogs behaviour. I find it facinating. Personally I have never seen any alpha dogs. I have never seen any "dominant" dog going around pinning other dogs to display its dominance. Most of the play is cooperative. If one dog plays to rough there's usually a yelp and play stops for a moment. Then usually resumes. If a dog doesn't want to be bothered they give a growl and the other dog walks away. There is a lot of subtle body language that make things go smoothly. Very occassionally there is a disagreement.
What I think I'm getting at is that dogs have their "rules" with each other. When I step in I don't think they see me as another dog. I'm pretty sure they don't. I have my own rules. I don't think of it as being dominant over them but I need some sort of control. For their safety.
As for my dogs they wait until I tell them they can come inside or outside. Is that me being dominant? Maybe. I like cleaning their muddy feet before they step all over the carpets. And most of the time I wish that people would teach their dogs to wait before rushing out the door! Imagine being pulled through the door by a few huge dogs that don't know how to walk on the leash! I think it is more about manners, safety, and just making things go a bit more smoothly.