If white color has a reflection close to one, what role does shininess or dullness play?
Just because two materials both reflect all of the light that strikes them doesn’t mean that they look the same. When you send a flashlight beam at a white surface, you can see that reflected light from all directions. When you send the flashlight beam at a mirror surface, you can only see the reflected light from one particular angle. Both the white surface and the mirror surface reflect virtually all of the light that hits them. A shiny white surface is different from a dull white surface because a shiny white surface has a small amount of mirror character to it: you can see the whiteness from any direction but there is also a mirror aspect that you can only see from certain angles.