I have browsed through a great number of sources for the definition, and here are several of them from reliable dictionaries:
Treccani: assegni su piazza, fuori piazza, pagabili rispettivam. nella località stessa dove ha sede la banca che li ha emessi o in località diversa.
De Mauro: fuori piazza ... di assegno, pagabile in località differente dalla sede della banca che lo emette.
Devoto-Oli: fuori piazza, di assegno, pagabile in una località diversa da quella in cui ha sede la banca che lo ha emesso,
and so on.
In the term database of the Government of Canada, the definition is provided for the English equivalent, out-of-town cheque:
A check [drawn] on a bank which is located outside the territory of the clearing house with which the collecting bank is identified.
My question is whether an assegno fuori piazza is a cheque drawn by a bank on a bank located definitely in another town (I mean that a bank in Rome draws a cheque on a bank in Venice, and therefore this cheque must be paid in Venice) or it is a cheque without the city stated (which means that the cheque can be paid anywhere and it becomes fuori piazza if I present it before a bank in Venice and su piazza if I present it before a bank in Rome).
UPD. To put it briefly, pagabile = can be paid or pagabile = must be paid in this case?