Yes that is my understanding as well, that he is speaking about the casualties of the initial assault on the 3rd being traditionally exaggerated. Of course those numbers of confederate casualties that you provided from the NPS have been questioned as well, for being too low.If I remember correctly, Rhea only questioned Grant's losses during assaults during a short period of time on June 3.
Campaign: Grant’s Overland Campaign (May-June 1864)
Date(s): May 31-June 12, 1864
Principal Commanders: Lt. Gen. Ulysses S. Grant and Maj. Gen. George G. Meade [US]; Gen. Robert E. Lee [CS]
Forces Engaged: 170,000 total (US 108,000; CS 62,000)
Estimated Casualties: 15,500 total (US 13,000; CS 2,500)
Description: On May 31, Sheridan’s cavalry seized the vital crossroads of Old Cold Harbor. Early on June 1, relying heavily on their new repeating carbines and shallow entrenchments, Sheridan’s troopers threw back an attack by Confederate infantry. Confederate reinforcements arrived from Richmond and from the Totopotomoy Creek lines. Late on June 1, the Union VI and XVIII Corps reached Cold Harbor and assaulted the Confederate works with some success. By June 2, both armies were on the field, forming on a seven-mile front that extended from Bethesda Church to the Chickahominy River. At dawn June 3, the II and XVIII Corps, followed later by the IX Corps, assaulted along the Bethesda Church-Cold Harbor line and were slaughtered at all points. Grant commented in his memoirs that this was the only attack he wished he had never ordered. The armies confronted each other on these lines until the night of June 12, when Grant again advanced by his left flank, marching to James River. On June 14, the II Corps was ferried across the river at Wilcox’s Landing by transports. On June 15, the rest of the army began crossing on a 2,200-foot long pontoon bridge at Weyanoke. Abandoning the well-defended approaches to Richmond, Grant sought to shift his army quickly south of the river to threaten Petersburg.
Result(s): Confederate victory
Also, it's interesting that the 1864 battle is considered so bloody with about 18,000 total casualties (13000 US, 5000 CS) when the 1862 battle in the same location had 15,000 casualties (7000 US, 8000 CS).