I am not sure why vampirequeen is being given such a hard time over her terminology. We often say what we think (and the word "bitch" can feature in my vocabulary especially when I am very angry) and to refer to her stepchildren as " our" children may not be what some of you would do, but it shows the existence of a family unit in vpq's thinking. Yes, the woman in question is playing power games and I too would feel defensive of my DH in similar circumstances.
What to do? Depends on the outcome you both want. Take it on the chin, suffer in silence, have as calm a conversation as you can manage on the ground rules?
Whatever, it is a potentially explosive situation and if the children's mother wants to be annoying, the best remedy is not to give her the satisfaction of seeing you riled.