Great question! I'll take a stab at it:
What do you refer to someone who is without a job? Unemployed?
But what if they have enough $ that they don't need to work, independently wealthy? Rentier?
We are defining the person based on their employment status.
To label a child illegitimate or not is squeezing them into a category based on traditional marriage, religion, state, and marriage contracts. Still quite normal for many families, and quite foreign for many others.
Legitimacy dates back to the days when marriages were contracts and involved the changing of money, land, titles, and the blessings of the church or state. Children born inside this union were legitimate to the ends of the contract or in the eyes of the law. Children sired outside this union were illegitimate so far as collecting inheritance, titles, and having a legal standing in the family unit.
Today, even children born outside of a marriage have rights so in a sense, all children born today are legitimate, they are merely outside of a traditional marriage.
I don't suppose you have to call them anything different than you would call any other kid. They are son or daughter of so and so. Now, if you're trying to differentiate a child born by one member of a marriage that is outside the family, say for example a husband has a child with his secretary, then I think it sufficient to mention that this child was born outside of this marriage and was the product of the reltionship between said man (husband0 and woman (secretary).