Part of the Pack

I often wonder how my dogs perceive me.  Am I their friend, master, playmate, pack member or some combination?  Recently I found a new way to analyze this through our pack relationship. 

Dogs work in packs.  According to many dog trainers, an owner should attempt to establish themselves as the clear alpha member of the family pack.  My wife and I take that approach, to greater and lesser success, with our dogs.  For the most part it works.  The dogs appear to consider us one pack and we do have constant struggles with Mira (the female) for dominance of the pack. 

The dogs though seem to treat people and dogs differently with respect to packs.  Whenever a person approaches us, Mira will happily jump up and show affection in hope that she can sneak in a quick lick or two.  However she often will react negatively to other dogs which seek to approach us. 

Why the difference?  Do my dogs view themselves as different than us?  Why are they afraid of new dogs but not new people?  Do they consider other people beneath themselves?