You see, Kievan Rus equals Russia is the thing I've heard only from Russian officials. After the Mongol invasion different parts had very different paths in history.

Yes, and the people in the domain of the Czardom of Russia never recovered. One could make a point that the independence of the Cossack tribes contributed its share to the "de-imperialization" of the Dniepr region. A benefit the peoples in the Duchy of Moscow and later the Czardom of Russia never had.