Yes, and I think that, in some circles, that is still regarded as the 'correct terminology' - although I think that some reserve that term for very small ones (literally just for 'washing hands'), rather than much larger basins sometimes found in bathrooms these days.
Yes, that point has already been made. Contact with either side of the output of the transformer should not result in an appreciable shock.
However, as I mentioned, if a significant amount of water were to get into the back box, it could, at least theoretically, partially 'bypass' the transformer.
As I and others keep saying, it's really all a matter of common sense.
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