Lip licking, yawning and sprawling out are all signs that generally indicate the dog is relaxed or happy.
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This, so what that says to your dog, when you yawn, is that you are happy he may approach you safely for comfort/companionship.
I wonder if he normally feels that you are unapproachable, the leader, the god, she-who-must-be-obeyed. I think that if my dog was doing this every time I yawned when sitting on the couch, I would try to be a little more relaxed and PLAY more with the dog. Training is great, but, I don't know, it sounds like maybe the dog is not aways comfortable in your presence. Maybe training would sit on a back burner for me, and I would spend more time yawning and scritching the belly, doing some tug, maybe do some agility.
Of course I could be all wet, and you may just have a dog with excellent manners.
Years ago, I had a friend over who was a little afraid of my dog. The dog in turn was a little wary of her. I told her to yawn. She did. The dog came up and they were buddies then. She was really shocked but it worked.