I will try and explain this from my point of view...
Letting a dog be a dog from, my standpoint, means taking the reins and learning to become a leader in life for your dog.
Dogs today have way too much pressure put on them to make decisions about life.
Our dogs need us to take the pressure off of them.
If you do some research on different breeds you will see that humans have selected certain traits along the line that would be most beneficial to the human (this is very evident in the working breeds - cattle dogs, hunting dogs etc.).
This means that the dogs have been bred to follow us and assist in whatever we bred them for.
If this job is not fulfilled, it will literally drive them crazy!!! (Ever wonder why doodles always steal things?...They HAVE to use their mouths for something, they are a mix of two water retrieving breeds that use their mouths to get the pray.) (Ever see a dog just running back and forth along the fence barking???...crazy or lack of a job?)
This means that the definition of letting a dog be a dog is for us to be the leader, make the hard decisions and let them follow.
The best place your dog can be is relaxed and happy.
Having a balanced dog means you need to fulfill all of your dogs needs.
The only way to do this is to understand what those needs are and to understand that their needs differ from ours.
I have actually put up a course in my online school called Wellness Plan for Dog Owners that covers every area of your dogs life that needs to be addressed and how to address it.
If anyone is interested, you can click this link: lorianne.teachable.com
If you are ever in doubt as to whether or not your dog is just being a dog, look at how relaxed they are.