My position:
'Do you believe God exists' is not the same question as 'does God exist'. To the first, I answer no. To the second, I answer yes.
This is more a personal difference than anything else. When people say 'I believe in God' they mean something completely different than 'God'. When you say 'God exists' it forces people to ask 'What is God?'. Believing in God does not. If they believe in God, they can believe that means just about anything. If God exists, then God has to be something, rather than anything.
This doesn't make any sense.
The use of the word "God" in "Do you believe God exists" is exactly the same use as in "does God exist?"
You assert that one is specific while the other is general but that isn't the case; I'm not sure where you get the bizarre notion that the use of the word "God" is different in those two contexts.
If you answer "yes" to the question "Does God exist" then it means you do, in fact, believe that a god exists. Nothing about the question indicates that you know anything specific (I'm not sure where you get this notion), just that a god exists. It's still a belief statement and it's still theism to answer affirmatively to the question "Does god exist."
So, indeed, if Roman Catholics answer "yes" to the question "Does God exist?" then they are theists. I'm still not quite sure why it's taken us several pages of text to establish this elementary fact, but I'm glad it's over.