The cylindrical part of a hinge is called a knuckle.
In some areas it is also called the barrel.
A knuckle.
hinge
Hinge joints include a cylindrical bone end that fits into a groove, called a trochlea, on the other bone. One example is your elbow.
It got its name because it acts as a hinge, only opening and closing a certain distance. Your elbow is a hinge joint. Think of your elbow as a hinge on a door in your house. The door is your forearm and the door frame is your upper arm/body.
The pin that slides into a door hinge is called a hinge pin. It runs through the hinge knuckles and allows the door to pivot smoothly as it opens and closes. Hinge pins are common in both residential doors and automobile doors, and they can be removed or replaced for maintenance, alignment, or repair.
The theorem emulates the action of a hinge. As the angle of the hinge is increased the distance between the free edges increases.
Cylindrical.
rouleau
Don't know what a "bushy" is but maybe you're thinking about a bushing. A bushing on a car is usually a cylindrical "sleeve" that's used if you want to attach one part to another but still allowing them to move a little. Think of it as a hinge lined with rubber. A regular hinge would seize and jam unless the halves are perfectly aligned, but with a rubber sleeve in one half, the parts have some degree of freedom to move.
The butt end of a hinge is the part that attaches to the stationary frame or door jamb. It typically has screw holes for mounting.
Synovial pivotal hinge
It is called a door hinge spring and can be bought at LMC for less than $10.