Congress doesn’t have to formally declare war. The USC says congress has the power to ‘make war.’ There is no constitutional requirement that congress votes and passes a bill called a ‘declaration of war’ prior to the US military being engaged. Their mandate to ‘make war’ can be met by passing a military budget, for example. To the extent the War Powers act holds the president or congress to a higher standard than the USC, a case can be made that the War Powers Act is unconstitutional.
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.