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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.

In the UK, at least, all taps are taps.

So my question is, what are indoor taps really called? Like this one:
http://biltema.no/no/Bygg/VVS/Baderom/Blandebatterier/Servantbatteri-86568/

That seems to depend on which English you use. The example in the photo provided, would be called a tap, in the UK. It seems that in the USA, they are called a "faucet".

tap

UK (US faucet) http://dictionary.cambridge.org/dictionary/british/tap_4

tap1

[= faucet American English] http://www.ldoceonline.com/dictionary/tap_1

It's called a faucet. I know, because John Lajoie was angry at his neighbor for fixing his¹!

It's good to note what the Wikipedia article says:

In the British Isles and most of the Commonwealth, the word "tap" is used for any everyday type of valve, particularly the fittings that control water supply to bathtubs and sinks. In the U.S., the word is more often used for beer taps, cut-in connections, or wiretapping. "Spigot" or "faucet" are more often used to refer to water valves, although this sense of "tap" is not uncommon, and the term "tap water" is the standard name for water from the faucet. Between "spigot" and "faucet", the connotative distinction is outdoor-versus-indoor, and utilitarian-versus-decorative; thus a spigot is an outdoor tap such as the bibcock (sillcock, hose bibb) for a garden hose, whereas a faucet is an indoor tap such as on the kitchen sink, bathroom sink, or bathtub, which usually include decorative features such as styling cues and polished chrome plating.

These words all refer to a traditional, manually operated valve, fitted into the bunghole of a container such as a barrel or cask: spigot, stopcock, turncock, and tap.

All valves that release municipally supplied running water are called taps, and such water is called tap water (even in regions where the word faucet is otherwise popular). These valves are never called stopcocks.

The words faucet and spigot are used in the USA. The distinction is mainly that faucets are valves which are stylish and decorative, whereas spigots just look like termination points for plumbing. The indoor vs. outdoor distinction is linked to this one, because indoor taps are attractively styled, whereas outdoor ones are not. This, of course, isn't always the case. Outoor style taps (de facto spigots) may be found indoors in laundry rooms, janitorial closets or underground parkades. The term outdoor faucet is not unheard of, but bathroom and kitchen fixtures aren't called spigots.

According to Wikipedia, the possible answer is that an outdoor water tap is referred to as a "spigot." I find this to be true in the Midwestern US where I reside.

http://en.wikipedia.org/wiki/Tap_(valve)

I'm in the U.S., and, while I'd be more likely to call my kitchen's water source a faucet, I would certainly understand tap. Moreover, there are instances where the word tap might be used, depending on the context:

This water tastes cold – did you get it from the fridge?
No, it's straight from the tap.