Overall and partial MOSFET voltage gain
I remember my MOSFETs class would often have a DC voltage source that was connected to the gate and drain of the MOSFET. The gate resistors would form a divider (R_G1 and R_G2) and bias the circuit, just like the equations you have listed. I believe you are getting information from a few sources that have different circuits (maybe a class and online?). I will answer your questions assuming what you're doing is similar to my college classes...

1) What is the difference between overall and partial gain when talking about MOSFETs?

2) What do each one of them mean?

3) Are both used in practice when applying equations to real circuits?

1 &amp; 2) 
Partial gain is Vo/Vi (where Vi is the input at the gate). 
Overall gain is Vo/Vsig (where Vsig is your signal BEFORE the blocking capacitor.

3) Depends on the circuit. Most people will not use MOSFETs for amplifiers and will instead buy an op amp or other IC, at least in automotive circuits. If you go to work somewhere that designs analog circuits, it will probably be more common, but I can't say for sure.