Ok, here is what happened in the original duel.
-The first BEWD attacked Beta the Magnet Warrior (defense mode) and destroyed it. This stays the same, either Yugi plays the REBD or not.
-The second BEWD attacked the Dark Magician. Yugi activated the Magic Cylinder, which directed the attack to Kaiba's Lord of D (defense mode), since, due to the latter's effect, the dragons on Kaiba's field could not be targeted by traps. The Lord of D was destroyed. This, too, stays the same, REBD on the field or not.
-The third BEWD attacked the REBD. Yugi activated the Spellbinding Circle, which lowered the BEWD's ATK points to 2300. The REBD destroyed the BEWD. If Yugi hadn't summoned the REBD, that attack would have been directed to the Dark Magician. Yugi would have activated the Spellbinding Circle all the same and the Dark Magician would have destroyed the BEWD. The rest of the duel would have been exactly the same, since the REBD's presence made no difference in the rest of the duel.