In the video the work is also called 'warping' where a line/rope would go round a bit (bollard) on the dock and the crew would 'heave' the vessel to the bit to create the forward momentum (similar to heaving a ship to an anchor). Hauling generally refers to when something in hauled up.

On later vessels, once they are rigged, this job can be done using a general-purpose capstan, and of course modern vessels use thrusters.....

A canal enthusiast friend once showed me warping marks on a canal bridge. These are from the days when canal barges were horse drawn and the tow path went up over the bridge and the bridge itself was used to provide leverage and over the years the tow-ropes cut into the stonework of the bridges.