I'm new to this whole process of obtaining Italian citizenship via jure sanguinis.
We are going this through my spouse's grand father.
There are some name and date discrepancies in the documents.
For example, he was born as Odoardo. Came to USA as Eduardo (ship manifest). He got married as Eduard. Naturalized as Charles (no mentioning of Eduardo)

On his son's (who's my spouses father) birth certificate he first was mentioned as Eduardo, then somebody crossed his name out and wrote Charles above it.
Also on his Italian birth certificate, his day of birth is 25. On Naturalization certificate day of the month is 24. On death certificate - 26.
My question is, do we need to change these discrepancies? If so how do we do it? I would be grateful for a link or a message that would explain the whole process of changing vital records step-by-step (in Connecticut).
Our consulate is NYC.
Thank you very much in advance for your help, it will be very much appreciated.
HudsonValley.