How does your partner react to all of this and how do they want to discipline their child? I would think that would be the best yardstick. Do what they do, as long as they are ok with it. Your partner won't have any respect if you do all the disciplining. Instead they will be an angel to be walked over and you a demon to the child. Consistency is totally important, setting boundaries that are enforced equally well by both of you is your best bet.
If it were my kid I would put her to her room and shut the door (she is 2), for 2 minutes and then ask for an apology. If no, then back to the room etc.. Works a charm (unless she is totally out of it tired, in which case nothing works except gentleness and patience to calm her down). You can't hurt them doing this. We do a three step kind of thing - we ask politely if she would like to do it, then we tell her to do it, then we say do it or she is going to her room, finaly room it is. Unless it is an established rule like hitting in which case it is straight to the room, or we give one warning. If it is in the car or when we are out then it is "once we get home you will go to your room" (they don't forget but won't remind you). As long as you always follow through then it has weight. As soon as you start slipping they are all over it and it stops working. You need to get tough where it matters but fair. The start is tough but it gets easier. You could also use sitting alone in the car as a substitute. Otherwise ignore the bad behaviour like you did, or you could stop the car, hop out and leave them in there for a bit. Just make sure you tell them why you are doing it and what it is they are doing that is unacceptable. Once their time is up that is the end of it, they did their time, they apologised, and things can carry on.
Statements like "in our family we don't hit" are powerful messages.
I think the name is a secondary issue here. She is probably copying her other mums distaste for her ex when she uses the word daddy i.e. when mum uses it she is angry so I'll use it when I am angry with them. I think your partner and your ex need to come to some agreement or you will always have this problem.