What is defined as a basin fixed to a wall, countertop, or floor, with a drain and a supply of hot and cold water?

The correct answer is "lavatory." In plumbing and building terminology, a lavatory specifically refers to a basin that is permanently affixed to a wall, countertop, or floor, and is equipped with both a drain and a supply of hot and cold water. This defines the functional purpose of a lavatory, particularly in bathrooms where they are commonly utilized for washing hands and face.

While "sink," "washbasin," and "faucet" may seem related, they do not carry the same specific definition as lavatory. A "sink" can be a more general term that includes various types of basins, often used in kitchens or laundry rooms, and may not always imply a permanent installation. "Washbasin" is often used interchangeably with lavatory, but it may not always convey the same formal classification within building codes. A "faucet," on the other hand, refers specifically to the mechanism that controls the flow of water into a basin, and does not encompass the basin itself.

Understanding these distinctions clarifies why "lavatory" is the most accurate and specific term for a basin that fulfills the criteria outlined in the question.