The publication of Naomi Klein's No Logo at the turn of the century marked the time when much work was being put into branding and identity marks. As a fresh graduate of art-school, I too was thinking about logos and identity marks and defaulted to the standard artist self-portrait.
It was relevant to me that an artist is a person rather than a corporation, and since we indentify people by their faces, a self=portrait made the most sense, especially because it worked within the long tradition of artist self-portraits. This was before the concept of "selfy" developed, and social media and online avatars became common.
This is a branding mark I created in 2017 to distinguish my professional web-freelancing. It was used on invoices and business cards, and now obviously on the homepage.