Pic related from Assisi, near(ish) the Vicolo degli Illuminati; my guess for the ancestral home of the family. The mention of Florence that I saw in the cap from the article was about the Bargello, which was originally the palace of the podestà (a bit like a mayor).
I don't know how old the Podesta family is by name, but they were not among the court of the Medici or their familial enemies, so if they're from Florence then they're a newer group.