I haven't understood which part should be divided by pi if the light is not implicitly causing pi*1 light.

When normalized both lambert diffuse and blinn-phong NDF terms have Pi in the denominator so:

But that I don't understand MJP's comment, that the pi is important for balancing diffuse and specular. How so if they both are affected equally?

He's saying that if you remove the pi from the diffuse and leave it in the specular the lighting wont be balanced because the diffuse will too bright, and the same will happen the other way around, if you remove the pi from the specular and leave it in the diffuse the specular will be too bright.