Ok, so you are an Atheist but are not clear with the concept. I will try to explain.

Atheism is not the belief that God does not exist. Atheism is the lack of belief in a God. 

This is because the default position is to not believe anything that isn't proven to exist (cookie monster, talking butter, dragon in the basement, etc.).

Say, you say you believe God doesn't exist. This is a proposition you can never prove. It's impossible to prove something doesn't exist, because it's obvious it doesn't exist until it's proven to exist. It is therefore an illogical proposition. 

Atheism/t is the guy who gets told by the religious guy: There is a God! Pray to him!

The atheist says: Ok... wheres your proof? You have none? Well... screw that then. 

You were as much an atheist before the religious guy asked you the question as after. Before he asked, you didn't know anything about it, so you naturally didn't assume it's existance. After the proposition, you were not shown proof and your position therefore didn't change.

This is the same for any claim. The space cookie monster for example. Had you thought about him before? Did you think it was real before i brought it up? If i say he is and fail to present evidence, did your position change?

See, the burden of proof is in the person that makes the claim. The atheist makes no claim. He rejects the claim made by the religious guy. The "rejection" is the default state for everything. If it wasn't like this everyone would live in chaos and fear of everything and anything being real at any moment.

In the same way: Baby is born. He doesn't know what God is. This mean he doesn't have a belief in God. He is therefore Atheist. Maybe he gets shown the proposition later and decides to accept it, he becomes a theist, or not and mantains the position.

This is apparantly very shocking for our fellow religious buddy there, wich i find hilarious. Anyways, hope this helped you understand it. Feel free to ask if you still have questions. I'll try to answer to the best of my abilities.

Yes, atheism is the lack of belief in a god, which is why the terms atheist and agnostic are not mutually exclusive. I myself would be considered an agnostic atheist, though when asked I generally tend to refer myself to the latter as there are also theistic agnostics. But calling someone who does not know of the concept of a god an atheist doesn't make sense as they could begin to believe in one as soon as they become aware of the idea, and thus in terms of debates and discussion it shifts away from the central argument.

Atheism is about wether you believe the proposition or not, agnosticism is about wether you know the proposition to be true or not.

The agnostic says that he doesn't know enough to accept the proposition, the Atheist says he doesn't believe the proposition. They are the same thing. You are just saying you don't know and don't believe the proposition rather than just saying you don't believe the proposition. It's the same thing. It doesn't change the conclusion.

A theistic agnostic is a bundle of contradictory nonsense. You can't believe it and know it not to be true. Those are most commonly known as the closet Atheists.

I don't know if you catched my previous post but this page i linked before explains all you need to know.

Hmm, would it make more sense to you if i said the baby is an atheist but he doesn't know it? I am not sure why my previous example didn't click though. If i say not believing the space cow means you are a "atheist-moo" but i didn't tell you, would that mean you are not a "atheist-moo"? It is not dependant on you recognising the term yourself, it's about your belief of the proposition.

I'd say that yes, it does mean I'm not an atheist-moo, since I wouldn't even know of the concept.

Quit trying to redefine a word that is self-defining.

That would mean you didn't know you were one, not that you weren't. I see that the problem is you can't recognise the default state as the rejection. 

So, do you think anything can be real until it is presented to you?

Is that all you have to say? I don't make up words nor definitions. I use the existing ones. You have me confused with religious leaders.

I addressed this before, some people begin to believe in a deity as soon as they are made aware of it, so it can can also mean that someone didn't know that they weren't one, so again that's why knowledge of the concept is important.

How do you reject something that you're not aware of? If someone doesn't know that the Detroit Red Wings won the Stanley Cup in 1950, that doesn't mean that they're rejecting that they won it that year.

How do you reject something that you're not aware of? If someone doesn't know that the Detroit Red Wings won the Stanley Cup in 1950, that doesn't mean that they're rejecting that they won it that year.