Power supplies with or without 'COM' and without ground port at all
simulate this circuit &ndash; Schematic created using CircuitLab

Figure 1. Various options.

Photo 1


The first photo shows a PSU with configuration of Figure 1a. There are two isolated supplies - isolated from each other and from mains earth.
Normal use would be to connect 5V- to COM and now you would have a dual variable supply for the analog electronics - typically +/-12 V - and a 5 V supply for the digital logic.
If the circuit requires mains earth for any reason then connect the green post to the relevant point. Typically this is the COM.


Photo 2


This power supply has remote sense inputs. These allow the power supply to compensate for voltage drop in the wires to a remote load.
If not required then wire as shown in Figure 1b. Note the shorting links in your photo.
If remote sensing is required then open the links and wire as shown in Figure 1c. The voltage between the + and - terminals will vary with load but the voltage across LOAD2 should remain at the setpoint.
Again, if an earth reference is required then this can be achieved using the green post.








simulate this circuit

Photo 3

Figure 2. A dual supply can be used in multiple configurations.

This has two independent supplies but without the earth option. These can be used independently, as a symmetrical supply or, for example, as a +12 V and +5 V
supply. Note the connection (or lack of) between them in each case.



From the comments:


  So if i were to use a power supply with remote sense inputs, and if i went to use it like the first power supply with COM, i would connect what would've been connected to COM to the green post.


I think you are confused. The Photo 2 PSU has only one output with + and - terminals. It is not a dual supply as shown in Photo 1. You can think of it as a variable voltage battery with an optional earth connection. You always connect the load to red and black and add an optional link from either to the earth terminal.

Have a look at my answer to Actual electric potential at terminals of battery and it may help your understanding.