FWIW, our company has "Designers" on one team who do general graphic design stuff, used by Marketing, Sales, etc. along with web uses; and we have "UX Designers" on a separate team who focus more on application layouts, usability, UX, etc. -- at times using images and such from the "Designers", and sometimes designing their own graphics.
So, my guess is either you use two titles, or use a slash, e.g. "Graphic/Web Designer" or "Graphic/UX Designer"?