My US passport shows my state of birth only, my visa shows nothing, my PdiS shows the city of birth and my carta also shows city of birth.
When I got my first carta di residenza they had the birth date wrong so I went back a couple weeks later and showed them my PdiS (I have the card) with the correct date and they corrected it for free. I would suggest you show them you PdiS only for a correct carta di residenza. You do need the carta di residenza for getting your patente so unfortunately I think you will need to keep pushing at the anagrafe until you get the correct response.
When we applied for our residenza we had to have original birth cert and have it translated and appostilized (?) before they would process our residenza.
