Sounds like a dug well. We used to have a driven well, but I believe that my sister replaced it with a drilled well. A drilled well could reasonably be called a "bore", though I think I'd be more likely to say "bore hole". And I'd think a "bore hole" was an extraordinarily deep well, such as is drilled for oil, or by a scientific expedition.
I saw a dug well once -- and actually drank the water, being under the impression that it was a driven well, dug wells having gone obsolete before I was born. I felt sick when I found out. (I suppose dug wells can be clean, but this one pumped out *ants*.) (It belonged to city people living in an old farmhouse.)
(Must ask my older sisters: we may have gotten the driven well when we got electricity and indoor plumbing. I know from one of Father's stories that dug wells were common when he was young.)
Other American wells: our well was at the bottom of a "false well": a deep cylindrical hole that put all the pipes and valves below the frost line; the handle for turning on the hose was the bent top of a long rod that ended in a fork stuck into an ordinary valve handle way down there. I remember Dad putting a watermelon in the false well to cool it. The pump was in the cellar, why did they bother to run a pipe back into the well, instead of putting in a sillcock? (But the freezeproof sillcock hadn't yet been invented.)
And, come to think of it, nobody ever told me where the driven well was; I just assumed that it was associated with the false well.
I once stayed in a house that had a "dry well" for disposing of gray water. I think that is slightly illegal now, but I think that downspouts sometimes run into "dry wells".
And the "wishing well" is a lawn ornament intended to look like the old-fashioned dug well; it usually consists of a ring of masonry and a small roof on two posts, and may have a simulated winch and bucket. It's usually somewhat less than life size.
Joy Beeson