Air pump generally refers to the overall air compressor, the role of the air pump is to compress the atmosphere (or other gases), so that it has a great deal of energy, and therefore also known as an air compressor, is to provide compressed gas.
A cylinder, on the other hand, usually refers to the use of working parts inside the compressor, such as a piston that moves back and forth in the cylinder, creating pressure on the gas.
Or machinery that uses compressed air has its own working cylinder.
Strictly speaking, there are many forms of air pump construction, and reciprocating is just one of them. There are also screw-type and others.