how to binarize an image?
There is Otsu method to use variance when splitting.

Taking your own steps you should convert RGB to grayscale (simple mean like $(R+G+B)/3$, or use luminance grayscale).

The next step is to give some threshold, from Otsu or arbitrary given by your choice, and operating on grayscale image everything below (or equal) is 0, so it will be black, and greater than threshold is 1 (white).

Operations on other scheme like HSL is exactly taking luminance, but the full change RGB -> HSL is not needed because you discard color info.

Situation when values are out of range does not occur, so do not worry.