Why butane exist as a liquid in a lighter ?
Now, am I right to say that the vapour pressure is equal to the
  pressure of the butane gases in the small container, that's why they
  undergo a phase change immediately?


You seem to have answered your own question.

The pressure inside the lighter container is indeed the vapour pressure of the mixture of n-butane and iso-butane that it contains, at ambient pressure.

Here is the pressure-temperature dependence for iso-butane:



(Source)

The high vapour pressure inside means that when you open the valve, a gaseous mixture of both isomers will come streaming out and some of the liquid mixture will spontaneously evaporate to replace it.