Given the context of the question (I checked the link provided for reference. That was very helpful to include!), I asked a skilled tradesman with twenty years of work experience in residential construction. He isn't a plumber, but he does work inside and outside of new and existing residential edifices on a daily basis, in the U.S.A. He told me that he has worked in the Midwestern states and the Southwestern states, and has colleagues throughout the U.S.A., though none in any other English-speaking countries. This is what he told me:
Faucet is the correct term for an indoor water tap. If one wishes to be precise, and the distinction is relevant, you may wish to differentiate between hot water faucet and cold water faucet. The reason that I qualify that with "if the distinction is relevant" is because in the U.S.A., most indoor sinks have a single outlet i.e. tap or faucet, for water, which dispenses both hot and cold water, depending on how the user chooses to adjust the associated knob or knobs labelled for such. The entire assembly of faucet and hot-and-cold knobs are sold as a single unit, so it is sometimes relevant to observe these distinctions when specifying for purposes of construction contracts.
Spigot is the commonly understood term for an outdoor water tap. Outdoor water taps dispense unheated or "cold" water. Indoor water taps usually offer the option of both hot and cold running water, but not necessarily. In residential construction, the feature of hot and cold running water is the standard. It is optional for commercial construction, depending on code and preference.