The military exercise, which involved 18,000 American and allied troops, offers a window into how Army commanders are countering not just Russian troops and tanks, but also twisted truths. The exercise occurred as President Trump is sidling up to Moscow by bad-mouthing NATO, calling for Russia to be readmitted into the Group of 7 industrialized nations, and planning a summit meeting with President Vladimir V. Putin of Russia next month.

American commanders say they are tuning out Mr. Trump’s comments — strengthening ties to allied armies, increasing the number of troops and spies devoted to Russia, and embracing Defense Secretary Jim Mattis’s newest defense strategy that focuses more on potential threats from Russia and China and less on terrorism.

Over the past year, the United States and its NATO allies completed positioning about 4,500 soldiers in the three Baltic States and Poland, and have stationed several thousand other armored troops mostly in Eastern Europe as a deterrent to Russian aggression.