"Anonymous. The line drawn across the floor is on the diagnal"

That is the line of progression, not the line of dance. I don't see any instruction to cross the line of progression, I only see an instruction to cross the line of dance.

"and is about two inches wide."

Interesting assumption...

"The direction or position of the second step is determined by the first."

Yes. A 3/8 natural turn continues with the direction unchanged during the first step, so the second step will land as indicated across the line of dance. However, a quarter turn diverts the progression sideways over the first step, so the second step lands down the line of dance from the first.