If you are confused about your own title, imagine how your clients will be confused about what kind of services you offer.
I think you should stick to graphic and web designer; that says it all. And if you do web, it's even better to use "web designer" as well when introducing your services; some people aren't even sure what a graphic designer does, but there's no mistake when saying you're also a web designer!
Graphic designers are probably the most multidisciplinary kind of designer. It refers to pretty much anything that can be digitally painted, illustrated, the layouts, printing, typography, etc.
Visual designer: I think it's very confusing and not the kind of term people search for when looking for a print or web designer. It sounds more artsy and less technical, and could be anything referring to art, video, painting, meat dress designer, fabrics, etc. Even though there is small difference between graphic and visual designer, visual designers seem to focus more on specific elements of design. Graphic & web design is more general and content/project management oriented.
You should use a title that will attract the kind of jobs you want to do the most. For example: Digital Artist | Graphic & Web Designer.