Thanks for the clarification and sharing your experience, Livio. He is my 3XGGF, too. I'll keep digging in hopes of solving the mystery. He seems to be his father's first child, so his mother could have died and his father remarried. Hopefully his birth record is obtainable...

On my ancestor's death record (#128, bottom record) his mother's name is Liboria Salerno:
https://familysearch.org/pal:/MM9.3.1/TH-1942-35350-23621-6?cc=2120739&wc=M9W9-5SL:1681447322

However, her name is encircled and a notation at the bottom of the record seems to indicate his mother is actually ...

The Antenati site seems to be down as of yesterday - not just the Bari records. I was working on my ancestors' records on Friday and haven't been able to access them since. It could be protracted weekend maintenance or the site is moving. I couldn't find further information about what is happening ...