Well Emma told Henry to call her Emma in the conversation we are talking about. He wanted to call her mom, but she said just call me Emma. It is usually other people that refer to Regina as his mother. Henry usually calls her like She or even the Evil Queen though i think he has said my mom before, but not to her. I don’t think we have heard him call her anything to her face.
And Emma did all but say in True North that she wants Henry. She said to Hansel and Gretel’s father that the only reason she doesn’t have Henry is because she has no choice when they were talking at the garage. It seems pretty clear to me after that episode that she wants custody and wants him away from Regina.