Barcelona is deemed the "main" embarkation port and, although I don't know the actual numbers, way more people board there than in Rome. Consequently, the MDR menu rotation is based on Barcelona being the "first" day of the cruise so that's when you'll see the welcome aboard menu.
During the port stop in Rome (Civitavecchia, if I'm going to be fussy), the ship will be virtually a ghost town, as most people do get off "because Rome". The first couple of hours can be a bit chaotic, though, with people disembarking whether to go on excursions or at the end of their cruise.
Otherwise, you will barely notice it, apart from seeing the suitcases belonging to the Rome disembarkers in the corridors and being smugly aware that it's not your turn to get off... yet. 😂