Apparently neither. According to the municipality's website ( http://www.comune.foligno.pg.it/categorie/142 ), as translated by Google, this is a marshalling of two coats of arms. The Argent section would thus be the upper half of a shield divided per fess.
My Italian is theoretically non-existent, but applying my French and latin, I conclude that Foligno's arms were in fact combined from the arms of "il capitano del popolo" (captain of the people) Per fess Argent and Gules in chief a Cross Formy of the second and of the city Gules a Fleur de Lis Or. The captain of the people was a magistrate appointed by the non-noble elements of the city.
https://en.wikipedia.org/wiki/Capitano_del_popolo
The only problem with this explanation is that the most famous capitani of Foligno were of the Trinci family whose arms are:
Illustre famiglia, d'antica nobiltà. I Trinci furono, infatti, signori di Foligno col titolo di gonfalonieri e capitani del popolo dal 1305 al 1356, e con quello di vicari pontifici fino al 1439. Il primo ad usurpare la signoria della patria fu Nello, che fino dal 1303 era capo dei Guelfi di Foligno, ed a lui successero in quel dominio otto individui della sua famiglia, l'ultimo, dei quali fu Corrado III
https://en.wikipedia.org/wiki/Trinci