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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.