Is it a condition to remain in the same prayer area after Fajr prayer to attain the reward of Umrah mentioned in the Hadith, or does moving within the mosque not invalidate this reward?

There is no harm in moving from one prayer spot to another within the mosque, and this does not diminish the reward. The place of prayer is where the prayer was performed, and moving to another area in the mosque with the intention of awaiting prayer does not invalidate that.