Looking at your simple example of genders using the same process the article uses to enumerate the combination of gender and days:

Assuming Child 1 is a Boy:

Child 2 can be: Boy, Girl

Assuming Child 1 is a Girl:

Child 2 can be: Boy, Girl

Assuming Child 2 is a Boy:

Child 1 can be: Boy, Girl

Assuming Child 2 is a Girl:

Child 1 can be: Boy, Girl

Combining those gives us the following combinations:

Child 1 | Child 2

---------+---------

Boy | Boy

Boy | Girl

Girl | Boy

Girl | Girl

Boy | Boy

Girl | Boy

Boy | Girl

Girl | Girl

Clearly there is duplication in there we need to remove any exact duplications before we get to the 4 possibilities you listed.

Now looking at the problem in the article again, but using a 2 day week for brevity:

Assuming Child 1 is a Boy on Tues:

Child 2 can be: Boy on Mon, Boy on Tues, Girl on Mon, Girl on Tues

Assuming Child 2 is a Boy on Tues:

Child 1 can be: Boy on Mon, Boy on Tues, Girl on Mon, Girl on Tues

Combining those gives us the following combinations:

Child 1 | Child 2

--------------+-------------

Boy on Tues | Boy on Mon

Boy on Tues | Boy on Tues

Boy on Tues | Girl on Mon

Boy on Tues | Girl on Tues

Boy on Mon | Boy on Tues

Boy on Tues | Boy on Tues

Girl on Mon | Boy on Tues

Girl on Tues | Boy on Tues

In exactly the same way, removing exact duplications gives us 3/7.

There is no ordering over the Tuesdays. "The older child being born on Tuesday and the younger child being born on Tuesday" is exactly the same as "The younger child being born on Tuesday and the older child being born on Tuesday". Just like "The older child being a boy and the younger child being a boy" is the same as "The younger child being a boy and the older child being a boy".