In all animals, dominance is the first right to limited resources. There is no need to rewrite anything on this term or make it more complicated than a simple definition. It makes classifying an aggression "problem" easier and the plan to rectifying problems easier.
Keep in mind that dominance is designed to reduce aggression. The aggression part is the result of a conflict. Dominance aggression can and does manifest itself between species if the aggression is triggered over the first right of a limited resource.
In the case of dogs, and other social species the most dominant in the group also are the primary leaders. Leadership and dominance are closely related in social animals but are not the same. It is important to understand the difference for troubleshooting purposes.
Leadership is the decision maker for the group. This does not mean that another member of the group may not suggest an activity or otherwise initiate an activity, but ultimately the leader is the decider of when, what, and if activities are going to happen.
With canines this can include travel, hunting, defending from an outsider, accepting an outsider, playtime, and other interaction.
Conflict over leadership can also trigger aggression in canines. It is uncommon, to hear the term "leadership aggression" because it has historically been lumped into the category of "dominance aggression". These two go hand in hand and since, in my experience, both need to be addressed in order to reduce confusion and conflict in an aggressive dog they should remain lumped together as "dominance aggression" as long as a trainer understands the subcategories within that need to be addressed.
Examples of "dominance aggression" (first right to a resource):
- first right to human or other dog attention
- first right over food/water
- first right over a toy
- first right to a resting place
Right to "lead":
- aggression when pack/family tries to leave
- aggression when pack/family member tries to initiate affection
- aggression when pack/family tries to initiate play
- aggression when pack/family tries to interrupt an aggressive behavior
- aggression when pack/family reprimand a dog