Can we clear up the definitions of the words "atheism" and "agnosticism"?

Any suggestions on how to better word this question are encouraged.

My understanding of the words:

Atheism:

The simple lack of belief in a god. If your answer to the question, “Is there a god?”, is anything other than “yes”, you are an atheist. The root ‘theos’ means “god”, and the prefix ‘a-’ means “without”. Hence, an atheist is someone who is “without a god”. If you do not have a belief in a god, you are an atheist regardless of what else you believe. An atheist may be utterly certain that there is no god; an atheist may simply disbelieve in all of the proposed gods; an atheist may simply be someone who’s never thought about the existence of a god. Atheism in and of itself is not a belief; it is a lack of a belief. Atheists may in fact believe things in relation to gods, but atheism at its core is simply the lack of belief in a god.

Agnosticism:

The position that the existence of a god is unknown or unknowable. If you cannot make a claim of absolute certainty about the existence of god (either way), you are an agnostic. Someone who deeply believes in a god but still concedes that they cannot know for certain is an agnostic. Someone who claims that the existence of a god is by its very nature unknowable is an agnostic. An atheist who believes in no gods yet concedes that it cannot be known for certain whether one exists is an agnostic.

Agnosticism and atheism are not mutually exclusive. One cannot be just an agnostic, nor can one be just an atheist. Everyone, by the very definitions of the words, is either an agnostic or a gnostic. Everyone, by the very definitions of the words, is either an atheist or a theist.

If you disagree with any parts of these definitions, please, by all means, explain your contentions here.