Congrats on the KMG.
Actually, the grooved wheel is a pulley when it is on a motor, and a sheave when encased in a lifting device.
A pulley is a mechanical device ... one of the six simple machines. It is a wheel and axle ( another simple machine) that is used with a load carrying device (engineer's term for a rope or cable) to change the direction of force applied to a load. This can be in any form from a bare axle to a grooved wheel, or an assembly of wheels and axles to transfer the force. Each force transfer device in a system is called a pulley.
A sheave is the term for the wheel mounted on an axle and encased in a frame ( the frame is usually called a block). The word sheave comes from a root word meaning disk or wheel. It is used with a rope or wire to reverse the force needed to lift or move a load, thus the whole assembly is called a pulley because it is used to change the direction of force. The sheave may be merely a flat wheel or grooved. The groove in a sheave is usually flat bottomed or rounded. Using two or more pulley/block parts makes it a block and tackle. The parts of a lifting pulley are - Block, Axle, Sheave.
When mounted on a motor, it is called a belt and pulley system. The direction of force is changed from rotational energy to linear energy, and then back to rotational, with the force in a loop (belt). In the early days of use, the pulley and belt were flat. Eventually, shoulder disks were added to the pulley to help retain the flat belt. They were often still referred to as sheaves at that time. It didn't take long to invent the "V" belt and the grooved pulley became the standard shape. After the mid 1900's, the term pulley was used to differentiate between the "V" grooved wheel used on a rotational energy device, and the sheave/wheel used in a lifting pulley device. The parts in a belt and pulley system are Axles, Pulleys, Belt.