An early controversy revolved about the issue of the President's powers and the necessity of congressional action when hostilities are initiated against us rather than the Nation instituting armed conflict. The Bey of Tripoli, in the course of attempting to extort payment for not molesting United States shipping, declared war upon the United States, and a debate began whether Congress had to enact a formal declaration of war to create a legal status of war. President Jefferson sent a squadron of frigates to the Mediterranean to protect our ships but limited its mission to defense in the narrowest sense of the term. Attacked by a Tripolitan cruiser, one of the frigates subdued it, disarmed it, and, pursuant to instructions, released it. Jefferson in a message to Congress announced his actions as in compliance with constitutional limitations on his authority in the absence of a declaration of war. Hamilton espoused a different interpretation, contending that the Constitution vested in Congress the power to initiate war but that when another nation made war upon the United States we were already in a state of war and no declaration by Congress was needed.Congress thereafter enacted a statute authorizing the President to instruct the commanders of armed vessels of the United States to seize all vessels and goods of the Bey of Tripoli "and also to cause to be done all such other acts of precaution or hostility as the state of war will justify . . ." But no formal declaration of war was passed, Congress apparently accepting Hamilton's view.
Nonetheless, a plausible argument can be made that the Congressional resolution you quoted is the functional equivalent of a "declaration of war" even if those exact words were not used. Certainly it goes a long way towards satisfying the underlying rationale for a declaration of war, namely, that it should be the decision of Congress rather than a single individual (the President) to formally commit our nation to military hostilities.