There appears to be a single node tagged as [well_type=shadoof] according to taginfo.
Unless someone here knows of another synonym in use I think the most logical thing would be to give them something along the lines of man_made=water_well + pump_mechanism=shadoof. It looks like the convention is to tag the “bucket on a winch” style of well as pump=no but I think it might be close enough to work as a tag.
Ones that aren’t part of a well could then have man_made=pump instead.
This is a new tag though so someone might come up with a good reason to use something else instead.
Tagging them as man_made=water_well is obvious. pump=no is also correct, going by the description and examples for the tag. If the bucket is self-emptying after being pulled up and doesn’t have to be emptied by hand, pump=manual would also be possible.
The examples on the water well page include some that also use hand pulled buckets but have wheels or handles to assist with lowering/raising the bucket. That aspect is tagged with handle=*. This seems fairly similar to the mechanism used by shadoofs.
man_made=water_well + pump=no + handle=<?>
Not sure what the best handle=* value for shadoofs would be.
In the same vein, I agree that the best approach would be man_made=water_well; pump=no; handle=shadoof (I haven’t been aware of the term shadoof until now). If we reach a consensus about that, I volunteer to update the wiki (I don’t think it’s worth a full-scale proposal, but hey, be my guest if you disagree).