Sorry for not being clear. Yes, it is a BMD Index I was looking at. For some reason I thought the information on this was all that was recorded, and was beginning to think that my grandfather was illegitimate like you mentioned! I understand now though that there's most likely parent information available on the actual certificate I order from the GRO.
As for the FamilySearch BMD index, I'm pretty sure that the second you listed is correct in my situation. My father has Alzheimer's and can't remember when his dad was born, so I'm going off of the Social Security index that lists his birthdate as being in December. I feel pretty confidant that this is the correct one, since it also lists that he was issued his Social Security number in New York and died in Texas, which are facts I know to be true.
I'm going to order a birth certificate. I'm eager to see his father's name and his mother's maiden name!