Here are some definitions for you, based on my experience with these terms:
A linkage is a group of bodies (or links) connected together by joints.
A kinematic chain is a subset of linkages, specifically referring to linkages with rigid, ideal joints, and rigid links. Since the joints and links are rigid, this allows you to use geometry to relate the kinematics (position and velocity) of each link to every other link in the chain.
A mechanism is synonymous with mechanical machine and refers to any collection of parts that can perform a mechanical action or transformation. For example, a gearbox is a mechanism that converts (transforms) an input torque and speed to a different output torque and speed without the addition of energy. The relationship between the input and output mechanical variables is known and quantifiable.
A mechanical element is the smallest subdivision of a machine that is still useful for analysis (similar usage as in chemistry, if you consider what happens beyond the scale of atoms to be the domain of physics). For example, if you are considering the gearbox from the previous example, and are interested in the motion of each shaft and gear within the box, then the individual gears and shafts would be considered as mechanical elements. However, if you were analyzing the entire drive train of a car, it would be more convenient to consider the entire gearbox as a single mechanical element, ignoring the internal motion and focusing on its input-output characteristics as it interacts with the rest of the system.
Terminology like this can vary from place to place, so it is important to consider your audience. If you are writing an internal report or presentation for a group of engineers or in a course where the terminology is known to you, just use their terminology. Otherwise, it is important to include definitions for any terminology you might use.