Straight vs Null Modem Serial cable confusion
I have checked that as per the above image, pins are shorted like this: 1-1, 2-2,3-3, 4-4, 5-5 and so on. 
  
  Does that means my cable is a straight-through cable as required?


Yes. A straight-through cable connects pins as corresponding pairs. Pin 1 to pin 1, pin 2 to pin 2, and so on. "Extender" cables (having opposite sex connectors at the two ends) are usually wired liked that.

A null-modem, or cross-link cable usually has similar connectors at the two ends, and therefore outputs and inputs have to be swapped (crossed) so that an input at one end gets connected to an output at the other end.