"Descartes said "I think, therefore I am." So in that sense god does exist."

er ... nearly ...

What he actually said was that after he had stripped away all the things which he couldn't assert to be true with 100% surety, (i.e anything perceived by his senses) all that was left was his mind.

"I am; I think" Was the only thing he could not find grounds to doubt.

He does not argue that talking about the notion of himself makes him exist, he points out that at the end of his discourse, he is left with "I " I walk, I think, I disagree, I think, I don't think, I perceive wrongly, I trust nothing ... etc ...

In all cases, as long as I , then I exist.

This has no bearing on any external thing such as God.

He did try to go on to argue that God exists, but not on those grounds.

His thoughts and observations require the existence of "I" to occur.

"I think God exists" is not subjective to God but is still subjective to "I". Likewise with "I talk about God and therefore cause the idea of him to exist"