It says to declare war, not to “make war”.
You are correct that the form of this Congressional authorization need not say “this is a declaration of war” and many such authorizations have not done so - they use the language “authorization for the use of force” or other phrases.
But the power to declare war is Congress's, not the power to “make war”. Better read that Constitution again.
Yes, I was wrong and thinking of something else. But the rest of the analysis stands: congress need not have to formally declare war to meet constitutional mandate.