This is the standard definition, AFAIK. If the kind of natculture you describe exists, it wouldn't be described as a moiety system.
I have read that a natculture like that exists in Africa. I am sorry that I don't know the name of the people nor do I remember where I read it.
The main feature of moiety-systems AIUI is that the culture is divided into two "halves". Most social interactions are allowed only between persons in opposite moieties; most others are allowed only between persons in the same moiety.
Clearly another part of the definition is that marriage is one of those interactions that can occur only between people of opposite moieties.
The system I described shares those features with a moiety system, but obviously a person's mother and father must be in the same half, namely, the half opposite that person's own half. So, since marriage isn't one of the interactions that can occur only between persons of opposite halves, this system is not strictly speaking a moiety system.
Another relaxation or broadening of the "moiety" idea is to cultures which are divided into more than two subsets; say, four or eight. (In these systems one cannot marry someone from one's own subset.) I've seen people call those "moieties", and seen other people correct them, or at least warn their readers.