How do I identify which point is Live and which point is Neutral of a 2-pin plug?
A plug doesn't have polarity, it completely depends on the wiring of the socket. To check the socket all you need is this two dollar tester:  

  

It has a neon light inside which connects via a high resistance to the contact at the end. If you insert in in the live pin and touch the metal dingus on the back of the tester, there will flow a very small, safe, leakage current to ground which is enough to light the neon light. If you try it on the neutral it won't light because the neutral is at the same potential as ground.  

edit
Well, that's how the classical tester works. Cybergibbons points out that this one is probably a non-contact tester (because it doesn't go deep enough in the socket?). Anyway, those detect the electrical field emitted by the phase's voltage. They're battery operated to power the LED.