Yes, and the back and forth presumably comes from rapidly changing local accounts and local perspectives, and the lack of a uniform strategic position on the Mongols across Christendom. They had that unity of purpose with the Crusades, but after their defeats, there was not a similar position vis-a-vis the Mongols.
The Mongols also cut off what later became Russia from mainstream European civilization. The Kievan Rus was integrated enough to have Vladimir the Great’s granddaughter Anna of Kiev marry King Henry I of France. So actually Felipe Vi, the current King of Spain, is a direct descendant of Vladimir the Great and therefore of Rurik. But under the “Tatar Yoke” Russian culture became more authoritarian. Ivan the Terrible saw himself as a Genghis Khsn type more than as a European King.