"faucet" and "tap" are the same - "faucet" is American English and "tap" is British English. You are correct about "mixer" (sometimes said "mixer tap" in the UK, sometimes just "mixer") - it is a tap with hot and cold water running into it but just one spout for the water to come out of. Some mixers have separate hot and cold controls for you to turn, while others have a single control which you can turn to very hot, very cold or any position in between.

Thanks for your comment. It is really funny.

Interesting that Americans use faucet when the word comes from old Middle English - the fact that we use a more archaic word than the Brits is rather strange!