We got scared

I suspect we're discussing a distinction without a difference. That said, I am not convinced that early religion or lack thereof was a cause for extra-community conflict, the neolithic equivalent of war.

As I understand it (most of what I have learned on the subject is incidental and I claim zero expertise), neolithic religious artifacts fall into two significant social areas, birth, typified by worship of female fertility figures and death where some sort of ritual was conducted with the remains of the deceased.

Later gods would be required to explain the sun, moon, seasons, tides and essentially everything else. I expect that formal religion grew out of all these beliefs coalescing and somebody who was not involved in the food gathering process interpreting and taking spiritual control of the community. Perhaps elders or infirm, incapable of participating in the hunt became the first priests.

The prototype city-states and early "empires" in Mesopotamia were the home of Zoroastrianism, the first known monotheistic religion but most accounts indicate that it was not imposed on those communities that were incorporated into the Empire either by force or diplomacy. I think this indicates that expansionist fundamentalism and conversion by the sword would come much later and become a feature of Christianity and Islam. Until Constantine, the Romans were remarkably tolerant towards other people's religions and those apocryphal christian martyrs were, in Roman eyes, more what we would consider terrorists today.

After Rome collapsed, religion in the form of evangelical Christianity and later Islam expanded with zero tolerance for non-believers.