What are the differences between these types of air cycle machines?
Really not sure what you're asking, nor do I know what these terms you've found mean, so let's start with these drawings and see where we end up.

DC-9 Bleed air enters from the engine (or APU) at bottom left of this drawing and enters the primary heat exchanger. Then it enters the compressor in the ACM and then on to the secondary heat exchanger. From here it enters the turbine and this is the point that the heat loss occurs and cold air enters the water separator and then the cabin of the airplane. 

Note the anti-ice screen and the pack anti-ice valve which adds a little heat if the air exiting the turbine is too cold and is causing ice.

The water separator is a cone-shaped cage with a heavy cotton bag over it. This collects condensation in the air and dumps it on to the face of the secondary heat exchanger to cool and improve it's efficiency. 

Temperature in the cabin is controlled by the temperature control valve on the left side of the drawing.



737 This system is very similar to the DC-9 and really the main difference is that the pack anti-ice valve here is operated electrically, while on the DC-9 it is done mechanically with a pneumatically operated valve.



757/767 This is a much different system, many more electrical sensors and control. The pack cooling fan is an integral part of the ACM, where on the other two systems they are separate components. The water separators are much more advanced and don't require regular maintenance like on the other systems.



As you can see, these three systems are all basically the same with the only real difference being how they are controlled and some of the ancillary components.