Consider this very simple formulation of Anselm's first argument:
- (Premise) A maximally great being exists in my mind.
- (Premise) If x exists in my mind and not in reality, then x is not a maximally great being.
- Therefore, a maximally great being exists in my mind and reality.
- (Premise) A maximally great being exists in my room.
- (Premise) If x exists in my room and not in the rest of the universe, then x is not a maximally great being.
- Therefore, a maximally great being exists in my room and in the rest of the universe.
So, (1)-(3) feels odd in a way in which (4)-(6) does not. Why? I suppose it is that to exist in my mind does not appear to be parallel to existing in my room. To exist in my room is to exist and be located in my room. But to exist in my mind does not seem to be the same as to exist and be located in my mind. Still, this does show that there is a way of taking (1)-(3) that makes it be a decent argument. For there is a sense of "exists in my mind" that does make it be parallel to "exists in my room". If my mind occupies a region of space, this is particularly easy: to be in my mind could be like being in my room or in the core of the earth. But if my mind does not occupy a region of space, there can still be a parallel. According to Aquinas, God counts as spatially omnipresent by virtue of knowledge and power. Well, he can be present in my mind by virtue of knowledge and power. Indeed, Christians talk of the Holy Spirit dwelling in them.
So now we have a reading of (1)-(3) that makes it a perfectly fine argument, as long as one has reason to accept (1), say due to an experience of the Holy Spirit's dwelling in one's mind.
I doubt that this is actually Anselm's argument. The main textual evidence against it is that Anselm takes God's existing in mind to be simply a matter of having the concept of God. I don't know how conclusive this is against the above reading. Anselm distinguishes between having a concept and being able to use the relevant words. It could turn out that in the case of God, to have the concept of God requires that God dwell in the mind. But all in all, as I said, I doubt this is Anselm's argument.
That said, the above reading does show that there are ways of reading (1)-(3) that make it be a fine argument. And I think it also highlights another point that not enough, perhaps, has been made of. The more existing in my mind is a genuine way of existing, like existing in my room, the better the argument sounds. But we know that at least the somewhat later medievals were drawn to views of mind on which to know an essence is to have that essence genuinely in one's mind, informing one's mind. On such views of mind, the argument may well be plausible, assuming one has reason to affirm (1) (for instance, because there is a presumption that if one has mastered C-talk, then one has the concept C).