The logo is pretty much telling you everything

The two places people are gonna see it is your site and if you hand out business cards, both instances theyre gonna know from the site or you telling them in person respectively . Google should index whatever is in your bio and metadata anyway which will include that so if they come through google, they'll see it.

"Web design" or any label is just one extra unnecessary element and looks a little amateurish. Not that youre an amateur, it just reads as atiquated. Nobody really does that anymore since its unnecessary. You can put that on the site elsewhere or on the back of the card as more of a description of services rendered, but i don't think it belongs in the logo.