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@jumblie I am not sure where does the up/downhill simile come from, so I am not able to elaborate on this too much. Maybe it has something to do with what pilot feels during skidding/slipping turn? See images in my answer. In the slipping turn resulting force felt in cabin pulls you toward turn center, which can be seen as "downhill" if you think angle of your seat relative to the horizon. During skidding turn net force points partially outwards, which could be imagined, analogically, as uphill? Could it be so?
@jumblie IMO has any aircraft in skid/slip to fly partially sideways. Approximately like you have sketched it in your drawing. Reason becomes clear when you look on forces from outside. You need, in addition to the lift some aerodynamic force pushing aircraft to the side. Reasoning will be little different for fixed-wing and helicopter (because rotor teeter allowing lift to point somewhat aslant relative to helicopter's fuselage -- someone actually flying helicopter can correct me here if I am wrong), but the result should be same. I'll try to add some explanation into my answer eventually.
One could tend to believe slip gives "more turning" because we have sideslips, right? But note that sideslip (or forward-slip, it is the same maneuver aerodynamically anyway, only ground track differs) is no turning at all, it is airplane flying straight forward, only difference is the nose points "wrong way".
@jumblie I believe, it is just other way around. Slipping turn gives (with the same bank angle) larger turn radius than coordinated one, skidding turn smaller. If it is not clear from images above, you can try to imagine it like this: would you fly according to the "step on the ball" rule, then in slipping turn the ball is "inside turn" and you have to step on bottom ruder to get coordinated turn. That is you need to "turn more".