Personally I would leave the cities as they are. It may not be totally historically accurate the have Samarkand and Bukhara as Khwarazmian cities in the very beginning, but it will be much more historically accurate as the scenario progresses, because by the time the Mongols invaded Khwarazm, Samarkand and Bukhara were the most important cities in the empire. It would be hard to manipulate the Khwarazmian a.i. to capture those cities in time for the Mongol arrival.
The Mongol cities (Tatars, Kereits) other than Karakorum are not actually cities but names of Mongol tribes. Other tribe "cities" names that you can place are Merkits, which should be right under Lake Baikal (the big crecent shaped lake above Karakorum), Unggirat, Ongut, Oirat. I'll try to find a map that can show the areas were the various tribes dwelled.