"Away down in Belden Green... The whole earth shook in a quiver; Every devil had done his best To outrun the rest To get back to Washington to shelter." After the Union defeat, Abe Lincoln laments the cost of the battle
The First Battle of Bull Run did indeed end in a Union rout. This was, however, an oddly minor result. The Union army was made up mostly of volunteers called up for only ninety days of service; they were greener than grass, and not really able to fight, but the politicians forced Irvin McDowell to lead his troops into battle before their enlistments expired.
The Confederates, equally green, had the advantage of the defensive, and so were able to hold on. The Union army retreated, and the retreat became a rout, with soldiers streaming back to Washington. But the Confederates, as disorganized by victory as the Union troops were by defeat (and badly disposed; Beauregard's staff was so bad that more than half his troops were acting in response to orders Beauregard had thought meant something else), were unable to pursue.
Cox does not recognize the second of these texts, which he calls "War Song," as the same as the first. It seems clear to me, however, that they are.
The confusion comes in the first line. Cox's "Bull Run" begins
Away down in Beldon Green, where the like was never seen
The whole earth shook in a quiver.
The "War Song" starts
Down in Bowling Green, such a sight was never seen,
The earth all stood in a quiver.
The temptation, of course, is to associate the latter piece with a battle of Bowling Green (Kentucky). But there was no battle of Bowling Green. In 1861, the Confederate forces of Leonidas Polk moved into that part of Kentucky, and Albert Sidney Johnston had his headquarters there in late 1861 and early 1862, but Johnston's position was weak and he retreated without battle after Fort Donelson fell.
Braxton Bragg's 1862 invasion of Kentucky never moved as far west as Bowling Green. After that, except for a few minor cavalry raids, the Confederates never came close to Kentucky.
The only reasonable supposition is that "Bowling Green" is an error for the "Belden Green" of "Bull Run," or perhaps that "Bowling Green" refers not to a town but to an actual bowling green. - RBW