You're both right.I don't know Catalan, but I think it's easy to make a logical guess as to why the shortened version ("Barça") is written with a cedilla. In "Barcelona" there is an "e" after a "c". This changes to an "a" in "Barça". The sound of an "a" after a "c" must be different to that of an "e" after a "c" (which happens in Spanish also), so in order to maintain the original "c" sound in "Barcelona", the cedilla is required. I'm sure a Catalan will tell us if this is correct or not, but I'm pretty sure that must be the reason why.
Hi, I am catalan. Let me explain how Barcelona is spelled and shortened.There is no doubt that when Barcelona is shortened it is spelt Barça with the cedilla but is there a Catalan out there who could confirm if you ever spell the city Barçelona with the cedilla en Català