As a nation state, Italy has emerged only in 1871. Until then the country was politically divided into a large number of independant cities, provinces and islands. The currently available evidences point out to a dominant Etruscan, Greek and Roman cultural influence on today's Italians.
There are 2 Faeto. One is a proper town and the other is a neighborhood for another town. Go to www.nonsolocap.it and put in Faeto on top, click and when zip code for the Faeto name and province intials and regions comes up click arrow on the Faeto name(blue) and up will come the website for your town for both adminstrative and tourist information.
Normally in small towns in Italy because land is scarce, cemetaries do not grow beyond their original size and headstones are few and far between. Bones are taken out of the walls and ground after 20 years and put into a comunal pits and the process starts all over again. Names are not visibile save for death records. =Peter=
I assume they cannot find her birth record so that the next best thing is to see if they have aCERTIFICATO DI STATO DI FAMIGLIA STORICO which is a family certificate showing the date and place o of birth of each family member. The town may or maynot have kept such a record back then since it is inthe name of Leonardo Agriesti (estimate of birth is good for Leonardo 1870-1875.)
Perhaps if you write to the Uffico di Anagrafe (censimento=census) for Faeto they may have a record of the family in Faeto. Send a form letter to the Uffico di Anagrafe in Faeto at the same address that you sent a letter to the Ufficio di Stato Civile.
You can also check with the Archivio di Stato di Foggia "UFFICIO DI CENSIMENTI" if you wish to double check on Faeto records? By the way do you know there is a frazione "Faeto" for the town of Serramazzoni in the province of Modena in Emilia Romagna region?
As you know a frazione is like a neighborhood for the town of Serramazzoni. =Peter=