What's the name of the part of the dentist's chair used for pouring water into a glass so the patient can wash his/her mouth?
It's called a Cuspidor. The entire unit together with the facet and the sink.
I'm a dentist :-) so I know.
A slightly specialized water dispenser? Maybe call it a "dentist's water dispenser"?
You seem to be asking about the faucet and not the basin, so that's what I suggest.
I have heard my dentist call it a spout, as in "Now you can fill that cup with water from the spout."