We had our second appointment with our vet behaviorist this morning. We watched a few videos we took, and she explained Skye's body language. Like a dog showing her belly could be a sign of submission and not a request for a belly rub. She told us we did a good job and were on the right path (good positive reinforcement for the humans ;) Skye has indeed become more relaxed with Peter, and I started to be more aware when she was just tolerating our interactions.
Our plan is to continue on the training path as it is showing results. We are going to continue associating the word "yes" with rewards. As she is starting to offer some behavior like laying down and briefly glancing at me, we are going to reward these too. Skye also started venturing in our bedroom, so we are going to reward this with a jackpot (several treats at the same time).
When giving her treats, we are going to drop them a bit farther so she can decide if she wants to come back or not. This is part of giving her control on her environment. I will also pause after interacting with her to see if she wants to continue or not. Basically, I will be asking her for her consent.
She also gave us a game to play with Skye: hide treats in a room and get her searching for them. "Dogs can be motivated more by the anticipation of the food than by the food itself." - Karen Pryor.
I am feeling much more optimistic after this appointment as I see we are getting somewhere, even if it's going to be a long slow path. Like being okay when Skye decides to hangout in the living room behind a chair in a corner.