Which signals does a 3-pin mouse switch allow to differentiate?
I am wondering, which signals does a 3-pin mouse switch send


It is a passive device and it doesn't send signals; it can be regarded as an open circuit (high impedance) or, when activated, a short circuit (low resistance). This means that signals can pass through the switch or, get blocked by the switch.


  or is it able to differentiate between 3 states


Usually (mainly unless explicitly designed) it has a normally open contact, a normally closed contact and a common contact. When unpressed the normally closed contact makes a short circuit with the common pin and the normally open pin is open circuit. When the button is pressed, the roles swap over.


  The 3 pins are labelled "COM TERMINAL", "NO TERMINAL", and "DUMMY
  TERMINAL".


It's possible that the dummy terminal does nothing other than provide some mechanical strength when soldered or fixed into place in the target circuit. This dummy terminal would likely replace the normally closed contact.


  The metal spring can also be in between the two connectors and not
  connect to any of them, but float in mid-air (indicated by the dashed
  yellow line).


Usually this is a transitory situation lasting only milli seconds and cannot be held in this state due to mechanical hysteresis i.e. once the switch starts to move, it flips over fairly rapidly.

Some switches are therefore described as "break before make" or "make before break" and this describes how the contacts connect and disconnect as the switch operates. For instance, make before break devices will connect COMMON to NORMALLY OPEN before breaking with NORMALLY CLOSED.


  Can the switch differentiate between all three of these states


The switch is a passive unintelligent device and therefore if you want to know what state it is in you use electronic circuits to make that differentiation. The middle state (half way) is rarely ever used for anything meaningful except in electronic piano keys where the time taken to break then make is used as a measure of how hard a finger has hit a key.

UPDATE

Think of the switch as able to perform this type of analogous fluid flow function: -




Water can pass from the common port to the right hand port when the handle is in the position shown. Look at the red arrow
When the handle is moved to the opposite position, water can flow following the black arrow.


Back to electricity; the arrows represent flow of electrical current.