After looking at the text of the Constitution, for me the next most important places to look for an understanding of war powers -- or any other constitutional provision -- are the other authoritative American legal documents of the Framing period. I have in mind the Articles of Confederation, which acted as the existing national constitution at the time of the Constitution's drafting and ratification, and the state constitutions, which were the next most important constitutional documents that created state governments that were in many ways more powerful than the national government at the time.
I think careful consideration of these provisions will show that the power to "declare war" could not mean the power to authorize, commence, or start military hostilities. Here is the provision on warmaking from Article IX of the Articles of Confederation:
The united States in congress assembled, shall have the sole and exclusive right and power of determining on peace and war, except in the cases mentioned in the sixth article . . .
Again, as with my earlier argument on Article I, Section 10 of the Constitution, Article IX is revealing because it does not use the phrase "declare war" to mean start or begin military hostilities. If the Framers had wanted to grant Congress all of the power to decide on war, they would have used the same phrase as Article IX: "Congress shall have the sole and exclusive right and power of determining on peace and war." Also notice that Article IX uses the words "sole and exclusive," meaning to me that no other institution has any say in making war. Again, our Constitution of 1787 does not use that language -- as it does, for example, with the trial of impeachment by the Senate, which the Supreme Court has indeed read to exclude any other branch from participating.
Instead, the Framers of the Constitution of 1787 gave Congress the power to "declare war," which is a narrower power. Again, reading the document as a whole, the Articles of Confederation show that declaring war is thought of as a different power than making war. Article VI of the Articles of Confederation, which Article IX cites, further declares:
No State shall engage in any war without the consent of the united States in congress assembled, unless such State be actually invaded by enemies, or shall have received certain advice of a resolution being formed by some nation of Indians to invade such State, and the danger is so imminent as not to admit of a delay till the united States in congress assembled can be consulted; nor shall any State grant commissions to any ships or vessels of war, nor letters of marque or reprisal, except it be after a declaration of war by the united States in congress assembled, and then only against the kingdom or State and the subjects thereof, against which war has been so declared, and under such regulations as shall be established by the united States in congress assembled, unless such State be infested by pirates, in which case vessels of war may be fitted out for that occasion, and kept so long as the danger shall continue, or until the united States in congress assembled shall determine otherwise.
Article VI here shows that declaring war is about changing a legal status between nations, one necessary so that certain other steps, such as issuing a letter of marque and reprisal, could occur. But notice that Article VI does not limit states in responding to hostilities with other types of military force until after declaration of war.