Early in church history, monastic houses (contemplative) were not organized into a religious order of any kind, and they were chiefly distinguished by the rule they followed. Each community adapted the rule it chose to suit it’s own purposes, so there could be considerable variation in how this worked on a practical level.
Thus, Augustinians followed the rule of St. Augustine, and Benedictines followed the rule of St. Benedict (Benedicts’ rule is presumed to be largely the work of “the Master” with additions by St. Benedict, the Master is often identified with St. John Cassian). The common name did not imply any other connection. Benedictines are still somewhat internally self governing, although in very recent centuries they have been gathered into a sort of umbrella grouping under the Popes.
In the east, and in some parts of the west, most monastic houses adopted the rule of St. Basil.
In the early church, monastic houses were under the supervision of the local bishop, in much the same way a parish would be. If a new community budded off from an older one, it would fall under the jurisdiction of the local bishop at the new site and sometimes this was far enough away to be a different bishop. The new house would start out with the same customs as the older “mother” house so one would see a lot of “families” of communities which happened to share much in common, they identified with one another and cooperated a great deal but were very much independent of each other.
Later in the west, some houses began to establish subordinate houses, like “branches” which were controlled from the earlier institution, and a type of religious order came into being. The Carthusians and Cistercians are an example of this development, I think Cluny did as well. Eventually, this model of centralized administration was applied to the cathedral Canons in the west and adopted by new active religious orders like the Franciscans and Dominicans. These organizations were able to operate distinctly within the dioceses in which they settled and worked, not necessarily at the behest of the bishop.
In the east, no such developments took place, and the earlier model of monasticism under the direct supervision of the bishop prevailed. In places like Poland, Hungary and elsewhere, after the various local Orthodox churches were incorporated into the Catholic church under the Pope, the monastic congregations (following the Rule of St. Basil) and their properties were removed from the control of the local bishops and organized into the western model religious order.
That is the origin of the Basilian Order which is being discussed here. Later, the Vatican decided to place the Basilian Order operating in Eastern Europe under the control of the Jesuit Order, and it was reformed from a contemplative into a more active role. This meant they were more suited to parish assignments, mission work and teaching along the lines one might expect of the Jesuit community.
It also meant that they were more like a western religious order than eastern monks. In a way, the evolution of these Basilians symbolizes the changes in the Byzantine Catholic churches over time. One should note that the Byzantine Catholic churches are now slowly recovering their ancient traditions and practices, and the Basilian Order, still organized along western lines, seems to be gradually attempting the same.