But then, why aren't my ice-cubes made in the regular fridge tray clear? The same principle should apply: they should be clear at the top, where the tray is open.
Ice in a tray freezes from all sides since it's not insulated. And indeed, the ice from your tray is crystal clear on the sides, with the cloudy ice at the center of the cube.
In an insulated vessel, the sides and bottom remain liquid until the frozen portion grows down from the top, pushing the cloudy bit to the bottom, which you can then just lop off.
I've seen some other ideas like freezing water in a camping cooler with its top open.