Italian "di" == "of"; "da" == "from". Think Leonardo was "from" the small Tuscan town of Vinci. I can't speak for 14th century Padua, but by mid 15th century Florence people with any social standing were often Name di Fathersname Familyname. Andrea di Giovanni Bardi, for instance, is the name of a real person in the 1427 census ("catasto (http://www.stg.brown.edu/projects/catasto/)") of Florence. You can tell from her name that she was part of the household of Giovanni Bardi. Leonardo's full name was Leonardo di ser Piero da Vinci -- Leonardo, son of Mr Piero from Vinci.

As I said, I can't speak authoritatively for 13th century Padua. A little googling and Wikipedia browsing, however, suggests that Marchetto da Padova (http://en.wikipedia.org/wiki/Marchetto_da_Padova) was the name of an early Trecento (http://en.wikipedia.org/wiki/Trecento) composer, born in late 13th century Padua. Wikipedia is obviously not an official source for heralds, but at least it hints that you're on the right track.