There are many ways to approach this question. There are different aspects of these characteristics in terms of importance; they could be important to archaeologists, in helping us interpret the remains of the civilization that we find or they could be important in terms of helping the society grow into complexity. As for growing into complexity I believe all of the characteristics, including the secondary characteristics, can be fundamental to any society depending on the climate they live in and the general socio-political direction they are tending to grow in. Taking all of this into consideration, in my opinion, the most important primary characteristics for both measures of importance discussed above would be state authority or a state system of decision making and ability to enforce decisions.
I realize that there are also other characteristics that others may consider more important and fundamental for the growing complexity of a society however state authority is what I think defines a state as a state. Tribes or chiefdoms don’t have this level of authority in their government by cause of their general tendency toward egalitarian characteristics. Once we are able to identify that a civilization had this characteristic we are able to confidently confirm that the civilization in question was indeed a state and there is no ambiguity unlike the primary characteristic of agriculture or specialization where as these could be applied to both chiefdoms and possibly complex tribes. This characteristic is also relatively easy to spot in the form of state art and architecture. Since large works of art imply a strong state authority that can successfully command many people (slaves or citizens alike) into doing things that they want them to do.
However states don’t necessarily need state art to confirm that they indeed held the characteristic of state authority. We can also see the presence of state authority in the characteristic of a complex economy. In order for a state to distribute its goods widely a central coordinator is necessary. It is important for a state to be able to record transactions and distribute incoming goods to the population. This would be impossible without some form of state authority large enough to manage all of the incoming and outgoing goods in a large society. This also implies that the state is large enough to hold a large number of people which means urbanism is involved, something that a tribe or chiefdom lacks.
In conclusion, the primary characteristic of state authority is important not only for the central management of the other primary characteristics but also for archaeologists to identify a state in the future.