live life vs live a life

Hi, I often bump into these two expressions: live life and live a life.
The differences that I've found (I might be wrong) are in the following;

1. If you want to speak about it generally, you just say live life. For example, living life to the fullest is difficult.
2. If you are talking about a specific life, you need a in front of life. For example, if you want to live a happy life, if you want to live a life that you love, if you want to live the life that Jesus did, (in this case, you need "the" instead of "a"....

I want to know if my understanding is correct.

The article usage is the most confusing one for a person like me !