It's a sad thing that children's emotional well being become casualties to arguments involving their parents. This situation at Walgreens, we didn't see the whole story. But I understand mom's frustration if she was in a public hospital all day with her baby concerned about her health. For the sake of her daughter, I would hope and pray that this type of exchange was just an anomaly triggered by the stresses of the day and not a regular occurrence.
In the other situation, dude was in the wrong. He had nothing to lose by letting the baby's mom put their daughter in the car. But instead he's being an ass and letting his friend or woman film it. De-escalate, cooperate and go on with the day. This became an ego battle between the parents. And as you pointed out, neither parent showed consideration about how it was making the child feel or how this argument was affecting her.