Let's get some definitions out of the way, because there's a degree of confusion between all of these, not helped by multiple entries in dictionaries - which only add to the confusion - or by manufacturers who in the past have gone for over-the-top descriptions of their products. So, in the context of this website, here are the definitions I've adopted:
A well is simply a deep hole or shaft dug or drilled to obtain water, whereas a bore is by definition far narrower.
A pump is a mechanical device for transferring water through a pipe or spout from its source to another location or container.
A tap is a combination of valve and spout which regulates the delivery of water, under pressure, at the end of a pipe. Some rather grand taps mounted within a solid cast iron column were often termed water pillars.
A standpipe is a large vertical pipe through which water is delivered under pressure.
A hydrant is a discharge pipe with a valve and spout at which water may be drawn under pressure from the water mains, typically for the purpose of extinguishing fires, washing down streets, or flushing out the water main. (Modern hydrants can also be tucked away below road level, of course, under cast iron covers.)
A fountain is a device that produces and contains an artificially created stream of water, under pressure. The term pillar fountain, used rather too broadly by some manufacturers in their catalogues, can reasonably be used if the jet of water is directed vertically upwards, under pressure.
Therefore, ornate municipal taps are still taps, no matter how ornate, and not pumps, because of the simple fact that they are connected to the water mains. But (if you're still with me, and not gone off into a total stupor), to add to the confusion, some old pumps have been converted to work off the water mains; these have therefore become taps!