In posts of months past you claimed there was no distinction between the God of the philosophers and the God of Abraham, Isaac, and Jacob; they're the same thing, if God can be called a thing at all; you asked for an argument that they were [not the same], if I am not mistaken. Here is my attempt to satisfy that request.
The God of the philosophers is immutable, as a result of his simplicity; this implies that he cannot be affected and respond to the goings on of the natural order, including us. Whatever happens in the natural order, God is [not] changed or affected in response to it. The God of Abraham, Isaac, and Jacob, however, does seem to be so affected, on any reasonable reading of the relevant religious texts: in Christianity, he enters into the world to provide a means of salvation from sin, which presupposes his consciousness of sin freely committed by created agents; in Judaism, I would guess, he talks to and responds to the prayers of prophets and great leaders, destroys civilizations because of their sins (which again is an instance of responding to occurrences in the natural order), etc. I won't talk about Islam because I don't know enough.
In short: the God of Abraham, Isaac, and Jacob seems to be affected in various ways and acts in response to goings-on in the natural order, whereas the God of the philosophers, by his very nature as immutable, cannot be so affected. The God of Abraham, Isaac, and Jacob offers a way of salvation because of human sin, and promises judgment in the future for those who don't repent; the God of the philosophers, on the other hand, cannot be said to do anything in response to what goes on in the natural order.
[. . .]
Your argument is this:
1. The God of the philosophers is ontologically simple, and therefore immutable: he cannot change, and so cannot be affected by anything that occurs in the created realm.
2. The God of the monotheistic religions is not immutable: he affects and is affected by goings-on in the created realm.
3. If there is a property P such that x has P but y does not, then x is not identical to y. (Contrapositive of the Indiscernibility of Identicals)
4. The God of the philosophers is not identical to the God of the monotheistic religions.
The argument is valid (correct in point of logical form) if 'God of the philosophers' means 'God as conceived by the philosophers' and 'God of the montheistic religions' means 'God as conceived within the monotheistic religions.' And I do think that is what you mean by the phrases in question. (Correct me if I am wrong.)
But whether or not the argument is valid, it is not probative because the first premise is false and the second is dubious.
Ad (1). Only some philosophers hold that God is ontologically simple; Alvin Plantinga is a prominent contemporary theist who does not. One cannot therefore build ontological simplicity into the definition of 'God of the philosophers.' As for immutability, some philosophers think of God as mutable, Charles Hartshorne, for example. So one cannot pack immutability into the definition either. And similarly for other attributes. For some, there are broadly logical limits on divine power, for others there are no limits on divine power. There are different views about the omni-attributes. There are different views about the divine modal status. There are different views about how the causa prima is related to the realm of secondary causes, etc.
The point is that 'God of the philosophers' does not pick out some one definite conception of God. There are many philosophical conceptions of God even within monotheism. There is no God of the philosophers if the phrase means 'God as conceived by the philosophers.' Premise (1) therefore rests on a false presupposition.
I read 'God of the philosophers' differently. What the phrase refers to is an approach to the divine reality, the approach by way of discursive reason applied to the data of experience, the approach exemplified by Aquinas in the Five Ways, for instance. Or the approach exemplified by Descartes in the theistic arguments of his Meditations on First Philosophy. The God of the philosophers, then, is God approached by way of discursive reason. It is essential to realize that what Aquinas, Descartes, and others were groping towards using their unaided discursive intellects was not a concept, an idea, an ens rationis, or anything merely immanent to their own thinking. It was nothing merely excogitated, or projected, or abtract, or merely immanent to their minds. It was, instead, the real concrete God, transcendent of the mind and independent of all modes of approach thereto.
My claim is that what the philosopher seeks to know by discursive reason is the same as what the mystic seeks to know by direct, albeit nonsensible, experience, and is the same as what the religionist seeks to contact by way of belief on the basis of revelation. They approach one and the same God, but in three different ways. To employ a crude analogy: if there are three routes up K2, it does not follow that there are three summits. There is and can be only one summit. Similarly, there is an can be only one God. Reason, mystical intuition, and faith are three routes to the same 'summit.'
Ad (2). It is certainly true that God is portrayed in many passages of the Bible as changing and thus as changeable. But it doesn't follow straightaway that the God of religion is changeable. For perhaps those passages can be taken in a merely figurative way and interpreted so as to be consistent with God's immutability. Just as one must distinguish between philosophical conceptions of God and God, one must distinguish between Biblical portrayals of God and God. The God of religion is God as approached via faith in revelation; but what exactly the content of revelation is is something to be worked out by hard theological work. The Bible does not supply its own theology. One cannot simply read it and know what it means. One has toreason about what one reads. But that is not to say that theology is philosophy. Theology accepts revelation as data; philosophy does not.
Consider Genesis 3, 8: "And they heard the sound of the Lord God walking in the garden in the cool of the day, and the man and his wife hid themselves from the presence of the Lord God among the trees of the garden." Obviously, this passage cannot be taken literally, for if so taken, one would have to say that God, a purely spiritual being, has feet. But if he was walking around on his feet, was he shod or not? And what was his shoe size? Were his toenails properly trimmed? How many corns and calluses did he have, if any? There must be answers to these questions and a thousand more if God was literally walking through the garden and making noise as he did so. And furthermore, he had to have physical eyes if Adam and Eve though they could hide from him behind trees.
Since we know that a purely spiritual being cannot have feet, and since we know that only a purely spiritual being could be the cause of the existence of the physical universe, we know that the passage in question cannot be taken literally. So what exactly the content of revelation is in Genesis and elsewhere is not easy to discern. But we can be sure that any portrayals of God that imply that he has physical attributes must be taken figuratively so as not to conflict with God's spiritual nature. It may well be, though I am not prepared to argue it in detail, that portrayals of God as mutable must also be taken figuratively. So I find your second premise doubtful.
So I persist in my view that the 'distinction' between the God of the philosophers and the God of the religionists is entirely bogus. In fact my view strikes me as self-evident if one construes the relevant phrases in my way. The God of the philosophers is the divine reality, if there is one, which is approached by discursive reason applied to the data of experience, with no use being made of the putative date of revelation. The God of the religionists is the divine reality, if there is one, that is approached via faith on the basis of revelation. Clearly, there can be only one divine reality. For if there were two, neither would be divine given that only an absolute reality can be divine and given that the divine is that than which no greater can be conceived. Since there can be only one divine reality, the God of the philosophers and the God of the religionists is the same.