articles - As the mother of a child or As a mother of a child

May I know which one of these is the correct answer.

1. As the mother of a child


2. As a mother of a child

I have seen both phrases used. However, a child only has one mother, therefore I believe the phrase "As the mother of a child " is correct. However, I have seen both phrases used in many newspaper articles. May I know the difference?

Similarly, these two phrases are also used. However, a football club only has one manager. Therefore, why do people use the second phrase.

1. He is the manager of a football club.


2. He is a manager of a football club.