No doubt you may be inclined to rely on your choice of historians for their opinions....the libraries and Google are full of them. However, for the purposes of consummation of war time issues, the United States government itself affixed the beginning on Lincoln's actions, with no transactional relationship to Charleston Harbor.
The US Supreme Court constitutionally neither declares the start of war nor its end.
So the ruling you cite was strictly for administrative purposes in a statute of limitations case, in which the Court itself admitted war could be said to have started and ended on many different dates.
But the fact remains that, just as WWII for America started with Japan's attack on Pearl Harbor, so also Civil War started for the Union at Fort Sumter.
However, one notable point the Supreme Court ruling does imply: despite Confederates' military assault, the Union committed no act of war at Fort Sumter!
By sharp contrast, from Day One the Confederacy committed many provocations for war and at Fort Sumter (similar to Pearl Harbor) the first genuine act of war.
Lincoln's call-up after Fort Sumter and his announcement of a blockade do not by themselves make a war, as such actions have happened in history without resulting in war.
That's because throughout history, nations can bluff or bluster short of war to improve their negotiating positions, or for example, in Kennedy's blockade of Cuba, to prevent Russian missiles from arriving there.
So, a call-up of troops, by itself, is not an act of war.
And a blockade of ports, by itself, is not an act of war.
War is war, meaning actual fighting between military forces, and that's what Confederates began in their assault on Union troops in Union Fort Sumter, just as Japanese began at Pearl Harbor.