The Quasi-War was just that, quasi, and thus does foreshadow a lot of modern U.S. military operations short of full-on war. The CAP essay seems to be suggesting that as early as the Adams administration, more limited actions seemed to require more limited authorizations.
In fact, though, the first war this country ever fought—with President Washington as commander-in-chief and preceding the Quasi War by six years—lacked a full declaration too. And that war was a full war, nothing quasi about it. Of course I highly recommend reading all about it in my book, linked above, but the point, for this post, is that when Washington made the statement that Amash has quoted, he was well into a war against a powerful confederation of Native nations, living mainly west of the Ohio River, led largely by Blue Jacket and Little Turtle. The confederation had enjoyed the biggest victory, by far, that indigenous North Americans would ever have over U.S. troops. In response, Washington had gotten Congress—not to declare war—but to authorize and appropriate funds for the first regular, professional U.S. Army and to send it, under General Wayne, affectionately known as Mad Anthony, to crush the confederation and conquer what was later to become known as the Midwest.
Context is everything, and in context, Washington’s words, quoted above, don’t represent some philosophical reflection on the constitutional limits on his own warmaking power. What was going on: There had been some thought of extending the U.S. war of invasion on a new front, against the Creeks in Georgia. That seemed inadvisable to the Cabinet. But a correspondent formerly in the South Carolina line of the Continental Army was eagerly lobbying Washington to approve organizing a bunch of Georgia militia to do their own thing against the Creeks. To squash that plan, Washington wrote to the guy and referred, with his usual perfectly controlled formalism, and in no uncertain terms, to an “offensive expedition against the refractory part of the Creek nation, whenever Congress should decide that measure to be proper and necessary” (my italics).
Then he made a blunt warning, which might as well have been underlined too, so I’ll quote it again, this time in context: “The Constitution vests the power of declaring war with Congress; therefore no offensive expedition of importance can be undertaken until after they shall have deliberated upon the subject, and authorized such a measure.”
Washington’s only purpose in making those remarks was to assert the sole, uniform warmaking powers of the federal government, as a whole, over the ambitions of motley, poorly organized and trained state militiamen, whose ill discipline he disdained, and who kept incorrigibly going off on their own and screwing things up. (Read the exchange—it’s classic GW.)