I agree that the behaviour you describe for Eze could be rooted in insecurity. I was thinking more of those dogs that run up to you and flop on their back inviting you to rub their bellies with no signs of submissiveness, only happy wagging
Typical in labs and cockers. I can see why you don't like what Eze does, but at least she is secure enough to expose her belly. That suggests to me that she is being careful, not fearful. My girl would never do that if she felt worried about greeting someone, she would back away, tail between her legs, maybe even give a bark depending on how stressed she was.