DT, I agree with your grammatical interpretation of "Our own Family." But the Rosatos were operating seemingly as a separate family only because they chose to do that. They bossed a crew within the Corleone family, but wanted more, were recalcitrant, and had the backing of a formidable OC Corleone opponent.

However (and it's a big one), the whole idea of one or more capos in a family being given permission by their Don to form another OC family is patently absurd.