I have never heard the term 'Rodin coil' but what you can see in the linked videos are normal electromagnets. The advantage of these Torodial coils (wikipedia) is that you can built transformers with almost no stray field. The magnetic flux is completely contained within the inside of the windings.
This is relatively easy to understand if you think about a long normal coil and then bend both ends together so that the whole coil is closed.
Such a coil can also be used to compare currents without changing or influencing the circuit too much. You probably have one at home as a Residual-current device to protect you from any current flow from a broken AC device:
The relay 1 cuts off the current flow if the currents through the L and N lines do not cancel each other, which means some other current path to ground.