One afternoon late August, as Russia’s military pursued its invasion in Ukraine, one of Russian President Vladimir Putin‘s top advisers was mustering public-relations officials from government bodies and ministries for a fight on the home front.

“The main war that is taking place right now is the war over people’s minds,” Sergei Kiriyenko, a longtime aide to Mr. Putin charged with key domestic and Ukraine policies, said in a speech at a closed meeting. “All of us in this room are the special forces fighting this war.”