You capitalize God when you are referring specifically to the Christian god, and you lowercase it when it's just, well, a noun. It's like the difference between Earth and earth and, as people have pointed out, the Sun and sun. It just is. That's the way they decided to do proper nouns in English. Christians use the word god as a name for their god, so you write it God. Everyone knows what it means and to write it another way is unnecessarily confusing.
Putting it in quotes is just wrong. Sorry! It's my pet peeve to see quotes where they don't belong. Now that I've been abrasive, I'll be nicer. The Christian god is not a nebulous concept. I mean, it's got a million definitions that no one can agree exactly on, but when people who believe in a god are talking about that god, it's pretty clear what they mean. If you are setting up emphasis to make a comparison that's one thing: "Some people say that "God" is the comfort of a light breeze..." or if you wanted to indicate a category: "For those who believe that "God" is a collection of all the nice things in life..." That would be correct. But just throwing quotes around something to show that you think it's fictitious really isn't appropriate. Whatever you are putting in that sentence will probably get your point across. It would be like saying: "Hamlet" was a crazy loon who wasn't nearly good enough for "Ophelia."
Extra quotation marks are almost always just clutter that look like mistakes. It's very, very hard to make pointed statements with alternate punctuation.
So blah blah blah, what I meant to say was in response to this:
So basically I'm wondering if "God" is really just the Christian name for their god, or if it's a "theft" of the word, which should properly refer to any god. I'm hoping it's the former because it's much easier to type and say "God" instead of "the Christian god."
I think God is just the Christians' name for their god. It's translated the same in the romance languages that I know. I don't know what you mean by thefting the word; my hunch--worth nothing, probably--is that the lowercase, common noun god followed the Christian proper noun God. Once people caught on that not every culture believed in their actual God but rather something analogous, they stripped the word of it's special status with an ordinary version.