I personally think too much is made of the question "What do you mean by God?". Yes, the concept needs to be defined, and there are so many definitions floating around that probably no one person has the same definition as the next. However, that's probably true about any definition when you get down to the nitty gritty of asking for 5,000 word essays of the topic.
The underlying definition of God that I think underlies all theist beliefs is that there is a metaphysical order to the world that is not accounted for in a completely random (i.e., no hypothetical God-like mind could discern a pattern) origin to the world. Now, some self-proclaimed atheists might be included in a theist classification in this sense, especially if they think 'mathematical patterns' exist and 'originated' before the universe, however they are not real atheists in my view. You see, the moment you start implicating metaphysical causes (even mathematical or logical ones) to the physical world, you have the problem of where do you stop in this process of implicating metaphysical causes. For example, in another post I showed where the Buddha, who supposedly was agnostic about God's existence, implicated a metaphysical cause to why one suffers in this life as due to law of karma which tracks their behavior from one life to the next. It's not 'God' per se that does this tracking, but c'mon, we can't seriously suggest that this is atheism since metaphyscial determinations are all around us in this case and dictating ordered sequence of events - destiny if you will. This qualifies as theism and not a true atheist belief.
Similarly, if you start implicating mathematical causes to the world (e.g., Alexander's position), then where do you stop? Why can't our lives be mathematical causes? Why can't the person we marry be a mathematical result, etc? Obviously there is no logical reason to say that mathematical causes is the reason there is the universe, and yet leave out this line of causes from influencing social and individual directions in life. It has to be considered part of theism.
If you will, metaphysical order is God. This is what separates a belief in Santa Claus from a belief in God. We are not attributing any metaphysical order to Santa Claus when we say there is a Santa Claus (unless we are serious), but we are when we attribute physical circumstances to this order.