There is not standardized names for these roles and a lot of overlap. Rather than use the labels, I would recommend you test for understanding:
"Designer": imagines how a product or service will look, behave, be constructed, or fits into user/customer's lives. Might not actually "implement" a design. For instance, a designer might work with artists to create the actual images or layouts, with programmers to code websites or apps to behave a certain way. Many implementers say that they are designers but their designs might not be very good. (
For an example you might be more familiar with, compare an architects (designs the over all building) and a carpenter (builds/implements according to architecture plans -- might be able to build without plans, but their buildings might not be as usable or attractive one designed by a talented architect, but that architect might never swing a hammer.
"Web Designer": "Designs" websites. May or may not implement them.
Design may focus on appearance ("graphical designer") or
interaction / behavior ("User Interaction (UI) designer") or
the totality of the user experience including both appearance and behavior and connection with the user's other activities ("User Experience (UE) designer").
Note that a UI or UE designer might specialize in design for non-web applications, such as native apps, physical design ("industrial designer"), etc.
I recommend you TEST for expertise by requesting the following:
Web Design: Show me a "specification" that you were given for a website. Show me the resulting site. Explain to me how your expertise was critical in converting to the specification to the resulting website and what that website achieved (measures of success).
Graphic Design: Show me the written specification of what your graphic design was supposed to achieve. Show me the resulting graphics you created. Why were these great choices?
User Interface / User Interaction design. Show me the specification for what users were supposed to be able to do using the product or service. Show me how you connected user goals to the interaction steps (number of pages, clicks, gestures, etc. they have to use to achieve their goals). Explain why this is a good set of steps for that purpose.
Find out how much the person implemented themselves, and how much they used other skilled experts. And pay close attention to how well they are able to explain the thinking of their designs.
The designer must be able to design for the engineers, the business stakeholders, other designers (current and future), and the people testing the software. I also need to be able to ask the engineers questions about the limitations of the software. For example, an engineer may have written a component that doesn’t care about another component. I may or may not have that knowledge and design something that creates a dependency between the two components. These differences must be spelled out so that everyone is on the same page before anyone touches the code.
Understanding limitations is important at the end of the process, but it hampers creativity in the beginning stages of design. Using the architect analogy, go look up architects' first drawings for actual buildings that have been built. Frank Gehry is famous for drawing these crazy scribbles that then eventually become things like the Walt Disney Concert Hall. If Mr. Gehry had been concerned about things like load-bearing walls during the first half of design, the building would probably not be as remarkable as it is. If your designer shows you scribbles in your first meeting, it's a good sign.
I hope that helps.