I don't even know what "the usual definitions of God" are, or why we should accept them. (It seems like an implicit demand that Christian theology be accepted a-priori.)
Well, classical theism is affirmed by Christian, Jewish, Muslim and Pagans thinkers such as Aquinas, Maimonides, Avicenna and Plotinus respectively. Classical theism affirms at least the following:
1) If God exists, God exists necessarily, in other words, God could not have not existed.
2) If God exists then He has no limitations, God is not limited in perfection, in power, in goodness and in knowledge.
3) If God exists then nothing can come into being or continue to happen without God creating it and sustaining it in existence.
There are various arguments (or proofs if you want) that attempt to establish the existence of God. Aquinas' 5 ways for example use change (1st), causality (2nd), contingency (3rd), degrees of being and transcendentals (4th) and final causality (5th) as premises in proofs that can in principle demonstrate the existence of something that just is necessary being itself (3rd), whose essence is its existence (2nd), that is intelligence analogously speaking (5th), that just is good (4th) and is purely actual (1st) and that is what classical theists call God. If such arguments are successful then it just logically follows that every contingent being that has ever existed and will ever exist is evidence for God.
The definition of 'God' that people are talking about here seems to be approaching a 'Matrix'-style 'brain in a vat' situation. That's where all of our possible experience is simply... fake... a simulation, not real. It's an illustration of the bigger and far more ancient problem of skepticism. How can we ever tell the difference between truth and falsity if every possible piece of evidence might itself be false? (To a certain extent, the whole history of modern post-medieval Western philosophy is a giant meditation on the problem of skepticism.)
I think you are right. And I think part of the problem is a mechanistic view of matter and a Humean view of causation.
Science provides us with a pretty good means of producing naturalistic accounts of natural events, by linking them together causally.
I don't see how this is a problem for theism really. Of course that depends on how you define "science" and what exactly you mean by "naturalistic" and "natural events". A theist may use science as simply to mean knowledge. Knowledge gained via our senses (empirical knowledge) and knowledge gained via intellectual abstraction all contribute to the theist's view of reality. To the theist a naturalistic account may mean that the existence of something that just is necessary being itself, whose essence is its existence, that is intelligence analogously speaking, that just is good and is purely actual is perfectly compatible with explaining and having naturalistic accounts of natural events. Causality for the theist may also be more complete as it include perhaps Aristotle's four causes, powers and the distinction between per se and per accidens causes etc. All these things may then form part of a pretty natural explanation of reality for the theist. Soit really depends how define your terms.
If a theist account doesn't add anything to that, then it becomes kind of superfluous from a physical point of view and can be dropped. That's what's happened over the last few hundred years as natural theology's receded.
Indeed. This appears to be accurate ever since Aristotelian metaphysics have been replaced with mechanistic views of reality.