Web Design vs. User Interface Design

According to Wikipedia, user interface design is the design of computers, appliances, machines, mobile communication devices, software applications, and websites with the focus on the user's experience and interaction. Utilizing the same source, web design is the skill of creating presentations of content that is delivered though the web.

I searched google to see if I could find anyone debating the two terms as they are used often interchangeably, but I don't necessarily agree that they are one in the same.

Our agency is a user interface design firm that utilizes a user-centered design approach to improve the user experience in a creative and conceptual way. Google searches done today are looking for a web design firm in Chicago, but are they really? Hmmm...