Americans use the term faucet to refer to what we Aussies call a tap. When we say tap, we mean the channelling tube out of which the liquid flows. Tap handles, or sometimes called spigots are turned to release the flow of fluid.
If I judge by the contexts in which I read the term, it seems that faucet refers to the whole box and dice, both of the spigots (the handles) and the tap (the channel).
Is this a correct impression?
Yes.
Thanks. 
Didn’t mean to come off snippy. But yes, us laypeople typically call the spigot and tap part as the faucet.
Disclaimer: YMMV, IANAPlumber, use only as directed by your licensed physician. 
Thanks.
I refer to the whole thing as the tap, handle and all.
Some Americans are faucet-centric in that they’ll refer to the whole device (tap with spigots) and the tap itself as a faucet, and the handles as ‘faucet handles’.
Most Americans will sometimes use the more specific but rarer disctinction between tap and spigot (pipe opening and its handle) and still call the whole thing a faucet.
But all Americans refer to the water that comes out as ‘tap water’ as opposed to ‘bottle water.’
And all Americans refer to the device which dispenses beer from a keg, whether directly or at the end of a tubing system as a ‘tap’ – such beers are ‘on tap.’
(As far as I know with my experience of Americans, being one myself.)
Peace.
We, okay I, will refer to the whole deal as the faucet, but more often call it a tap. The outside water distribution for the hose is always a tap and the faucets inside are frequently taps.
The handles, however, are usually handles. While I have used the term spigot it usually doesn’t appear in conversation. I’ll usually ask someone to ‘turn on the tap’ or in the case of my kids, ‘don’t touch the handles’. I equate a spigot to a simple gravity-fed drain mechanism like one you might find on a big coffee urn (despite the fact that apparently the industry calls them faucets…).
Depends on context, really, and most of the time it’s understood in context. However when you want to buy a nice spigot with valves, they’re called “faucets” in the stores, i.e., if you buy a “faucet” you receive a single unit that consists of the spigot and one or two valves or handles – nicer units are lever operated with a hot-cold mixer.
I don’t know if I’m weird or not, but I call the handles “valves.” Maybe I’m too much of an engineer? I don’t honestly know if I know if other people here call them handles, either – it either doesn’t come up or I just never thought about it. Just for the sake of it in Spanish they translate as “keys” – or at least my wife calls them “llaves.”
The outdoor thing? I think I call it either a faucet or a spigot or a valve. I think mostly valve, since the actual spigot is integrated with the valve.
Thinking of my wife again – she says the equivilent of “open” and “close” the water, whereas I tend to say “turn on” and “turn off” the water. Is there any regional difference in how we all say this?
In my experience, the word “tap” (relating to water, not beer) is a nonspecific reference to sources of “tap water” in general. A more common example would be “the store”:
Q: “Where did you buy that widget?”
A: “I bought it at the store?” (as opposed to from a guy or from eBay or from…)
“The store” doesn’t refer to any particular store. Instead, it refers to just the concept of a store.
Similarly, if I said: “This water tastes funny. Is it from the tap?” I would not be questioning the cleanliness of a particular tap (in the Aussie sense here). Instead, I would be questioning the cleanliness of water coming out of any nonspecific tap (in the Aussie sense here.)
The above sentence does not conjure up a mental image of any sink hardware at all. If I had said, “Is it from the faucet?” I would get a mental image of the specific faucet I had in mind (knobs and spout and base and all – the whole chrome unit.)
Regarding “spigot”: this may be regional, but I use “spigot” only when referring to something more… ummm… industrial than what you find on your kitchen sink. Like the thing of the side of the house that you connect your garden hose to – a spigot.
I call them all faucets, including those on the outside with the threads, which are really called hose bibbs.