We use 'tap' too, but mostly in very specific contexts. Tapwater is water from the faucet, as opposed to bottled water. And beer can come bottled or on tap, as in "Do you have Miller Light on tap?" I have also heard tap used to mean faucet in general, but 'faucet' is definitely more common.
However, it is true that no one in America says 'lift' or 'petrol.'
It's funny how subtle these cultural differences can be, and how something very commonplace can ring just a little bit false when a British writer is using American characters. For instance, there is one fic in which someone plugs in the kettle to boil. Americans don't use electric tea kettles; ours sit on top of the stove burner. But how on earth would one know that? The trouble goes the other way when American kids write Harry Potter fanfic.
My grandmother lived in India for years and there was a very confusing evening when she insisted that the maid should put napkins on the dinner table. The poor maid thought her misstress had gone bonkers, but she complied. (In India 'napkins' means diapers, whereas in America it means serviettes.)
