None of them do. Both Cox's initial report and Benning's report (commanding the defenders) say it was the movement of the Kentucky Light Battery to an enfilading position close to the bridge that was decisive:
Benning, commanding brigade:
"The next morning early (that of the 17th) the skirmishing was renewed. It continued, constantly growing heavier on the part of the enemy, till about 9 o'clock, when our skirmishers were driven in. At about 8 o'clock Captain Eubank discovered a large body of the enemy opposite to him in a wood within range of his guns. He opened fire on them and drove them in confusion from the wood, and with loss, to judge from the movement of their ambulances. Not long after his battery had finished this work it was ordered away. Thus the two regiments were left at the bridge without army artillery supports whatever. The general line of battle of our army was nearly, if not quite, three-quarters of a mile in their rear, and not a soldier was between them and that line. The intervening ground for a great part of the way was a long slope facing the enemy's batteries, and thus commanded by those batteries, so that re-enforcements, if they had been sent, would have been cut up by shells before they could have reached their destination. A regiment had been posted on the right further down the creek, but this soon after the battle commenced abandoned its post and went to the rear. Thus the two regiments were also without infantry supports, and without the expectation of receiving any re-enforcements. The two together numbered not more than 350 men and officers, the Second having only 97, and the Twentieth not more than 250. In their front was Burnside's whole corps of not fewer than 12,000 or 15,000 of the enemy's best men, with a numerous artillery. In this forlorn condition were the two regiments at about 9 o'clock, when the fight opened in earnest. At this time the enemy's infantry, aided by the fire of many pieces of artillery, advanced in heavy force to the attack; and soon the attack opened on our whole line as far up as the bridge. It was bold and persevering. The enemy came to the creek. The fire not only from their infantry, but from the artillery, was incessant, the artillery being so placed that it could fire over the heads of the infantry. It was met by a rapid, well-directed, and unflinching fire from our men, under which the enemy, after a vain struggle, broke and fell back. This attack was succeeded by two similar ones from apparently fresh bodies of troops, and with like results, the last of the two extending above the bridge to the upper part of our line. At length, toward 12 o'clock, the enemy made preparations for a still more formidable attack. A battery was placed in position from which it could command at almost an enfilade the whole face of the hill occupied by our troops. Soon it opened fire, and the infantry, in much heavier force than at any time before, extending far above as well as below the bridge, again advanced to the attack. The combined fire of infantry and artillery was terrific. It was, however, withstood by our men until their ammunition was quite exhausted, and until the enemy had got upon the bridge and were above and below it fording the creek. I then gave the order to fall back. Colonel Cumming, with two companies which had a few rounds of ammunition left, remained near the bridge as a little rear guard, and was, with these, the last to leave the ground. When he left it the enemy had crossed above and below him, and were coming up on both his flanks. They indeed cut off a few of his men by getting to his rear. The men of both regiments, though retreating different ways, were exposed for a long distance to the shells of the enemy. Under an order received from General Toombs they retired to a position near the right of the general line of battle. Thus at near 1 o'clock we were driven from the bridge, but we had held it long enough to enable the advance troops of General A. P. Hill to reach their position in the line of battle; and this, I suppose, was attaining the great object of defending a place so far in front of that line - a place so untenable as was the bridge."
Cumming, 20th Ga:
"On the 15th instant I was stationed at a post as picket guard, which post was the scene of action on the 17th, so far as my regiment was concerned. On the evening of the 16th instant, the enemy appearing in force, I sent out a body of skirmishers, who were driven back. Early the next morning the enemy showed in great force - at least three brigades, supported by artillery, among which were some mountain howitzers. They poured a heavy fire upon us, which continued almost uninterruptedly for nearly five hours. My regiment, about 200 strong, maintained its position, inflicting a heavy loss to the enemy, killing and wounding, according to statements of some members of my command who were taken prisoners, nearly 1,000. The fire of the enemy continued in undiminished force, and as all the ammunition available on the ground from the boxes of the dead and wounded men had been expended, and being enfiladed by a battery which the enemy had planted on our left, I deemed it proper to withdraw my command. This order was not given until the advancing enemy were within six or seven yards of my line."
Lewis, 2nd Ga:
"On the morning of the 14th [15th] we were ordered to take position on the banks of the Antietam River, to the right of the bridge. The Twentieth Georgia was on our left and directly in front and to the left of the bridge. We remained in this position, after having deployed the regiment behind trees and barricades made of fence rails, until the morning of the 16th [17th] when our pickets were driven in. The enemy commenced an attack upon the center of the regiment at 9 o'clock then immediately afterward upon the left, with how many regiments we are unable to ascertain. They were repulsed several times, but their re-enforcements continually came pouring in; and besides, all the while a battery completely enfilading our lines was playing upon us. We held the position until our last round of ammunition was exhausted. At this time Lient. Col. William R. Holmes, commanding, was killed, the command falling upon Maj. W. T. Harris. He, seeing the condition of affairs, ordered the regiment to retire by the left flank, which was done in good order."
There's a good reason why none of the principles mention Rodman's crossing - it happened after they fell back. Rodman's crossing was enabled when the 50th Georgia, trying to cover both fords in the area withdrew because Sturgis had forced the bridge.