Bright/Contrast node is described in both Shader Node and Compositing Node exactly the same way, word by word.
https://docs.blender.org/manual/en/latest/render/shader_nodes/color/bright_contrast.html
https://docs.blender.org/manual/en/latest/compositing/types/color/bright_contrast.html
Bright/Contrast in Shader Node Editor has no Convert Multuply checker, and, the most important thing:
Bright/Contrast is normalized in Shader Node Editor, while is a percentual value in Composting.
In addition to the documentation topic, I believe both nodes should have the same measure (normalized will be the best option in my opinion).
Regards