Here is a shot showing how lines are belayed on the replica brig Lady Washington. Bare belaying pins can be seen in the foreground.
Lines leading aloft are taken around the lower half of the pin and then criss-crossed in a figure eight pattern around the top and bottom of the pin to secure the line. The lines are not tied or knotted, the last 'turn' is wedged between the top of the pin rail and the first turn with a firm yank on the 'tail' end of the line. The free line is then coiled and secured to the top of the belyaing pin by taking the last foot or so of line (actual length depending on size of the coil) between the pin and coil and making a half twist which wraps through the coil and around the top of the pin.
To work a line, the half twist loop is removed from the pin and the coil 'capsized' on to the deck, so that the line will run free as it uncoils. The wraps on the pin are undone until a single 'S' shaped wrap remains with a crewmember maintaining tension on the 'tail'.
If the line is to be hauled, one or more crew members will pull on the line above the pin. The crewmember providing the 'tail' will haul the line through the 'S' wrap on the pin and maintain tension. The friction of the 'S' wrap prevents the line from pulling back. Final tension on the line is provided by a process called 'sweating'. When the line is as taught as pulling will provide, one or more crew members will pull together perpendicular to the line. Laws of trigonometry provide a huge amplification of force on the line. In choreographed 'dance' the crew on the upline side of the pin yank the line sideways and then down, gaining fractions of an inch at a time, while the member(s) on the tail haul the resulting slack through the pin.
If the line is to be 'eased', a crewmember 'tends' the line, feeding line into the pin as required. Lines may be traken off the pin to reduce the exertion on the crewmembers hauling on the other end of the yard, etc., but must be tended to prevent tangles or knots from fouling blocks. Removing a line from a pin is inherently dangerous though, comic as it may appear to others, getting a line wrapped around your ankle could spoil a perfectly good day.
In light of the above, belaying pins are not normally removed in line handling. Their lack of permanent attachment is for other practical reasons (moisture entrapment and resulting dry rot is one good one).
In the background, just past the cannon (which is not realistically rigged here) is a 'kevel', ( a board mounted across structural supports) used in securing heavy lines (in this case, normally the fore course sheet, but here it looks like it's being used with a dock line). Again the figure eight criss-cross is used to secure the line. Note that instead of a hanging coil, the larger line is 'flaked'on the deck. Flakes can be of various configurations, depending on the size of the line, space available and task at hand. Shown is a 'figure 8' flake. A rope coil on deck is a form of a flake.
To the upper left is a line secured to a cleat.
Hope this helps.
Schoonerbumm