Hold on, Back up. Everything was fine. The little girl did something wrong and got reprimanded. Mom hears what happened, and shows her child she agrees with you, by making her child apologize. Good stuff. Good role models.
Little girl throws a temper tantrum, and Mom says we're going home - which is very good parenting - very good indeed ! Those temper tantrums children pull are common and you need to nip them in the bud with consistency. The message is simple - dear child of mine, you throw a temper tantrum, there will be no second chance. Play time is over, you are going to your room.
You should have backed the other mother up by following her lead - helped her pack up and get the kid in the car. Instead of helping reinforce her rules, you undermined her. You sounded like you thought the mother was too extreme, as though you were sticking up for the bratty girl. Now that bratty little girl thinks you were on her side. The brat thinks she can behave that way with everyone but her mother. That mother was focused on her job, disciplining her child for bad behaviour and you were trying to discuss your parenting style with her.
Until kids are 7 you have to deal with all behaviour on the spot. No wait 'till we get home garbage. Every parent I know with really good, successful kids, has at least one story of a time they had to remove their kid from a restaurant, park, party, class etc..
You owe that mother an apology. Instead of helping her, you undermined her. Oddly enough, the other mother taught your son the same lesson she taught her child. Both children know that when that other mother says no, she means it (and you don't).
That little girl throwing a tantrum, in public, reflects her sense of security with her mother. When you reprimanded her, she felt bad, she knew it was wrong, but she didn't know what you would do and so she was scared. When her mother stepped in, the little girl felt safe again, and her anxiety and fear flew out of her in a temper tantrum. Her mother maintained control and made a quick exit - secure parents make children feel secure.
Stop psychoanalyzing little kids. Little kids need simple and consistent rules, and lots of safe space to play, experiment, try things out and learn (within the rules).
To me, you sound insecure - as though you needed the other mother's approval, acceptance or understanding. Stop looking for etiquette.
Parenting is not a group project, it is not a club, it is not a competition and it is not a Democracy. You and your husband are on your own. Your child will be the product of your choices, your decisions, your words, your boundaries, your examples, your care, your morals, your thinking and your actions.
