The distinction you're making is in your first example both the man is arbitrary, as well as the day of the week.
In the second example, only the man is arbitrary.
With those assumptions what you say is correct.
However, the question is "You meet a man on the street and he says, “I have two children and one is a son born on a Tuesday.”" and not "You meet a man on the street and he tells you he has two children, that one is a son and he tells you what day of the week he was born". So the day is not arbitrary, it's specifically Tuesday.
In the second example, only the man is arbitrary.
With those assumptions what you say is correct.
However, the question is "You meet a man on the street and he says, “I have two children and one is a son born on a Tuesday.”" and not "You meet a man on the street and he tells you he has two children, that one is a son and he tells you what day of the week he was born". So the day is not arbitrary, it's specifically Tuesday.