If I was conceived during a romantic vacation in Paris but was born some months later in Ohio, would it be right to say "my parents had me in Paris" or "my parents had me in Ohio"? Does "having" a child refer to the time of conception or that of birth?
In most English-speaking countries (I can't think of a counter-example, but who knows?), it is a taboo to discuss one's sexual activities and those of others in public. Childbirth, however, even though it usually is the result of sexual activity, is not a taboo at all.
Ergo, we celebrate our birthdays, and freely discuss when a woman gave birth to a child. When and how she became pregnant is normally not a topic that is widely discussed.
So when we say a woman had a child in January, we don't mean she conceived in January, but that she delivered a baby in January. "Having a baby" = "giving birth".