I have read Kaye's text that you've just quoted several times quite carefully, and I can't see the force of her objection. She is assuming that for Anselm for us to think about God is for God himself to be existing in our heads. In other words she's saying Anselm has gone wrong by assuming a principle like the following:
(1) If I am thinking of x having some feature F, then there is an object in my mind (called O) which is F.
Kaye points out, correctly, that (1) is clearly false. When I think of a rock, my mental representation of that rock is not itself a mineral.
Kaye seems to be saying that since (1) is clearly false, and Anselm's argument turns on (1), therefore, Anselm's argument doesn't get off the ground.
But I don't see why Anselm is committed to (1) at all. Anselm could deny (1) and just as easily say:
(1*) If I am thinking of x having some feature F, then I have some mental representation r that represents x's having F.
and then the argument could continue something like:
- (2) If I have a mental representation r that represents x's having F,
then either r corresponds to a real object (i.e. not a mental
representation) that has F or r does not correspond to a real object.
- (3) I am thinking of God as being the greatest conceivable being,
therefore I have an r that represents God as being the greatest
conceivable being. (by universal instantiation of 1*)
- (4) Therefore, either (i) r corresponds to a real object, or (ii) r
does not correspond to a real object. (modus ponens on 2, 3)
- (5) Suppose (ii). (Assumption for reductio)
- (6) If (ii), then I can conceive a being y who is greater than x.
- (7) Therefore, I can conceive of a being y who is greater than x.
(modus ponens 5, 6)
- (8) But I can't conceive of a being y who is greater than x.
- (9) Therefore, not-(ii). (5-8, reductio ad absurdum)
- (10) Therefore, (my mental representation of God) r corresponds to a
real object. (disjunction elimination on 4, 9)
- (11) Therefore, God exists (by 10).
That looks valid to me. Also, the premises look pretty plausible, as interpretations of Anselm. (2) doesn't turn on the specious move Kaye calls (1). (6) looks true because if x doesn't exist, then I can imagine a being y > x that is just like x except that it does exist. (That might not be true, but it's certainly something Anselm claims!) Finally (8) seems true by definition---assuming conceivability entails possibility, which Anselm does, you shouldn't be able to conceive of something greater than the greatest possible thing.