Which part of Phuket, Thailand, is the city center? I read many reviews about attractions but it is not clear where is the right place for a vacation. Is it 1 or 2?
The centre of Phuket town is the area you've circled 1. Your area 2 is Patong, which is primarily a tourist resort area - i.e. mostly bars, restaurants etc.
Which, if either, is the 'right' place depends on what you're after - e.g. If you're after loud music, parties etc then you're better off going to Patong, for a quiet beach go to one of the smaller beach villages such as Karon or Kata, or for the historic town stay in Phuket town itself.