Anselm's argument is addressed to someone who denies God's existence, and is intended to demonstrate that such a person's position is unsound. So far as I can see, it goes like this (my paraphrase):

1. You accept that God is defined to be that which cannot be improved upon
2. Even you, who deny God's existence, will acknowledge that it is possible to understand this concept of God
3. But if you say God does not exist, then the concept you have of God doesn't match (1), because what you're thinking of (something non-existent) has the potential to be improved upon (by being existent).
4. Therefore, if you can understand the concept of God in (1), you cannot help but accept that He exists

The conclusion feels uncomfortable, because it seems like we have been able to conjure God out of thin air simply by thinking about Him. And it does seem on the face of it that we have proved He exists.

But although the argument is characterised as a "proof of God's existence", I believe it is in fact a different proof, namely a "proof that it is impossible to believe that God does not exist". I do not think the two are quite the same: just because every rational person accepts that God exists, that does not mean that He does. (Although presumably, as a rational person, I am not in a position to argue that He does not exist.)

I think that this difference in emphasis is shown by Anselm's language. He says

• "For once granted that it exists, if only in the understanding, it can be thought of as existing in reality, and this is greater"
• "he who understands that God exists in this way cannot think that he does not exist".
• "something than which a greater cannot be thought exists so truly that it cannot even be thought not to exist"