Without knowing all the specifics I would say that "Domestic Name" is incorrect.
There are two probable situations that I can think of. The first being that she (like most Vietnamese) has her Family name, then her given name. What she was trying to say is that you used her family name, when her given name was more appropriate.
Or, you called her by a name that only her close family or husband may use. A nickname, or familiar name, that is rude for anyone else to use besides her husband or close family.
The general rules are that a Family name is passed on to children. All members of the same family share the family name (ignoring marriage)
A given name, is the individual name. This name is specific to each person.
A nickname is a name given to someone by someone else. Usually a joke, jest, affectionate, or personal name. Common ones that would not be acceptable to strangers and friends would be things like "baby", "cupcake", "honey", "love", "darling" etc. etc. Some times though these can be odd. They may even be part of a real name. I have a friend named Richard, his wife calls him Dick. Though a normal shortening of the name, he doesn't like it when others call him by that name.
Short version, Domestic name doesn't sound right. Unless she is trying to say that the name you used may only be used by her husband and children. Even then, it's an odd way to say it. She probably meant "Family name" or "private nickname".