I found recently that water fills a bottle quicker through only observation if I hold the tube away from the filling water than if I leave the tube inside the bottle. why does that happen? does potential energy have anything to do with this?
I'm not totally clear what exactly you are asking, but I think the answer is backpressure.
This "purifier" thing is apparently just a source emitting water out of the bottom end of a vertical tube. In your right drawing, the pressure at the bottom of the tube trying to push the water back up the tube is just the ambient air pressure.
In the left drawing, the top of the liquid in the tank is at ambient air pressure, so the pressure at the end of the tube is that plus the water pressure at the depth of the end of the tube. This higher pressure pushing back up the tube slows the water flow.