James Madison became the first President to sign a declaration of war, in 1812.
He listed the grievances he had against Britain and Congress decided to declare war. The British did plenty to provoke a war, but you say that it was the Americans who first decided to start fighting.
Thomas Jefferson , with permission from Congress, but without a formal declaration of war set ships and marines to the Barbary Coast against the Barbary pirates. I would not say that he started that war in any sense- the pirates were demanding payments for not attacking American shipping.
Woodrow Wilson was the US President during this entire war.
Truman at the start and Eisenhower at the end
woodrow wilson
George H. W. Bush , US President during the first Iraqi War .
President Mao
Franklin D Roosevelt was the US President at the beginning of WW2.
Franklin Roosevelt
Lyndon Johnson in 1966 was the first US President to visit Australia. He was seeking support for the Vietnam War.
Thomas Jefferson was the President during this undeclared war with the pirates of Tripoli ,also called the first Barbary War in 1801-1805.
Hannibal Hamlin
President Eisenhower.
Thomas Jefferson