God as an idea, a force, a wave, not a being or person or object.
An idea only "exists" within the mind(s) of conscious beings. Does your god only "exist" as an idea?
A force can be shown to exist otherwise it is just an idea.
I don't know what you mean by "a wave". A wave of what? Seems like you are trying to use scientific jargon to explain something. I.E., you are being metaphorical and vague.
Can you explain how you would test it?
Can we not look back though history and observe how the idea of God moves through people, influencing thought, culture and behavior?
Yep. That is exactly what a meme is.
From wiki: A meme is a unit of social information.[2] It is a relatively newly coined term and identifies ideas or beliefs that are transmitted from one person or group of people to another. The concept comes from an analogy: as genes transmit biological information, memes can be said to transmit idea and belief information.
A meme acts as a unit for carrying cultural ideas, symbols, or practices, which can be transmitted from one mind to another through writing, speech, gestures, rituals, or other imitable phenomena. Supporters of the concept regard memes as cultural analogues to genes, in that they self-replicate, mutate, and respond to selective pressures.
Advocates of the meme idea say that memes may evolve by natural selection, in a manner analogous to that of biological evolution. Memes do this through the processes of variation, mutation, competition, and inheritance, each of which influencing a meme's reproductive success.
Memes spread through the behaviors that they generate in their hosts. Memes that propagate less prolifically may become extinct, while others may survive, spread, and (for better or for worse) mutate. Memes that replicate the most effectively spread best. Some memes may replicate effectively even when they prove to be detrimental to the welfare of their hosts.[7]
Don't atheists point to the belief in God as the force behind the inquisition, the crucaids, modern terrorism.
No, its not that simple. Non-theists often point to particular worldviews held by the religious which stem from belief in specific gods, interpretations of holy books, and prophets.
You can't get from "I believe a god exists" to "therefore I should do X". It take more than the belief in some generic entity.
Don't 'people of the cloth' defend themselves with examples of how the idea or faith in God is the force behind charity, of the actions of saints, of social cohesion in every culture?
sure. Sometimes they are right and sometimes they are wrong. The religious often like to give their favored god(s) credit for all the goods things and none of the bad. E.G., if a train wreck kills 600 people then its just an unfortunate tragedy. But if a baby survives then its a miracle from god. Its quite silly the lack of critical thinking that occurs.
I would say that if God's influence can be observed, then God can be inferred to exist.
Sure you can. When one's imagination is the only limitation for explaining something then gods, pixies, demons, aliens, and a seemingly infinite amount of other things can be inferred to exist.
Once again, it goes back to whether you care if your beliefs are true. If you do care then more rigorous standards are applied than unbridled imagination.