It does not record a vote of the Confederate government for war.
The Act to which you refer is known as "An act recognizing the existence of war between the United States and the Confederate States, and concerning letters of marque, prizes, and prize goods."
Your date of May 6 is three weeks after President Lincoln started the war, as determined by the US Courts.
"The Act to which you refer is known as 'An act recognizing the existence of war between the United States and the Confederate States, and concerning letters of marque, prizes, and prize goods.' "
Only in a fantasy world could that be something other than a vote for war.
In the real world, the Confederacy prepared, provoked, started and formally declared -- "recognized" -- war on the United States months before a single Confederate soldier was killed in battle with any Union force, or a single Confederate state was "invaded" by any Union army.