What is a binary image and what is a $8-$bit image?

A binary image has $2$ gray$-$scale values $($black and white$)$ and an $8-$bit image has $8$ gray$-$scale values

Every digital image is a binary image and an $8-$bit image has $8$ gray$-$scale values

A binary image has $2$ gray$-$scale values $($black and white$)$ and an $8-$bit image has $256$ gray$-$scale values

Every digital image is a binary image and an $8-$bit image has $256$ gray$-$scale values