HELSINKI, Finland — In any other diplomatic setting, it would have been a funny, even insignificant gesture: One president handed a soccer ball, a symbol of teamwork and competition used in the most popular sport in the world, to another, just after the World Cup final.

But in this case, according to analysts and lawmakers who watched the Monday news conference between President Trump and President Vladimir V. Putin of Russia, the ball was a metaphor for the gamesmanship that Mr. Putin’s many critics say he is well-versed in — and for how he has scored points against an ill-prepared American leader in this high-stakes diplomatic exercise.

“The whole press conference was Trump repeatedly scoring own goals with the ball that Putin had given him,” Brian Taylor, the author of the book “The Code of Putinism” and a political-science professor at Syracuse University, said in an interview, describing a move in which a soccer player scores against his own team.

“Putin must’ve been extremely happy,” Mr. Taylor added.

The ball was just one of many dramatic moments during the joint news conference in Helsinki, Finland, in which Mr. Putin and Mr. Trump seemed to team up against the American press, Democrats and the intelligence community.