MOSCOW—In 1999, a series of apartment bombings in the Russian cities of Moscow, Buinaksk and Volgodonsk killed around 300 people and set off a national panic. Then-Prime Minister Vladimir Putin responded swiftly, overseeing the launch of an offensive in the breakaway republic of Chechnya and vowing to eliminate the Islamist fighters the Kremlin blamed for the attacks.

That take-no-prisoners response to terrorism propelled Mr. Putin, then a relatively little-known former intelligence officer, to the presidency. And it became...