Often, I hear people claim that they are 'Agnostic'. This is a misnomer, due to misunderstanding of Agnosticism. What such people mean to claim, is that they are weak atheists, or negative atheists, who lack belief in god. Such a description applies to babies, or people who have never heard of the idea of a god. Some would argue that it applies to people who simply don't know what they believe yet. However, not knowing implies a lack of conviction. A belief is a conviction in something's truth. If they lack that conviction in the existence of god, that means that they are at least an implicit atheist. Furthermore, that claim of simply lacking a belief in a god, does not take somewhat of a middle ground. The fact that they lack belief in all gods they know of, means that they have rejected all the gods possible, which is a position of strong atheism, or disbelief, at least of all gods they know of. Therefore, all people who have no specific belief in god are strong atheists, at least towards the deities they know of. Thus, the only people who are complete weak atheists are babies, or people with no knowledge of any supposed deities.
Agnostic, being an epistemological belief, does NOT refer to belief in god. As the coiner of the word, Thomas Huxley, emphasized, it deals with KNOWLEDGE of god, and is opposed to the "gnostics", who claimed knowledge of the divine(and also limited it to their initiates/acolytes). It is not a medium between Atheism and Theism. The problem with this misunderstanding of Agnosticism is mainly due to this common scenario. Somebody asks 'do you believe in god?' and receives the answer 'I'm agnostic', meant 'I don't know'. However, the original question was not about knowledge, but about belief. However, as to the question 'is there a god?' I can remain agnostic. I cannot know of every possible god, and thus cannot know that one does not exist. However, I still do not believe in that god, making me an atheist in respect to it. Should I be asked if I believe in a specific God, then I can say 'I'm an atheist', as I have my own reasons for believing that such deities do not exist, but furthermore, my atheism can be Gnostic(given I can provide evidence for why I lack such belief, and show how it contradicts itself, or creates a paradox/impossibility).
Agnosticism has two general meanings. One is that knowledge of the divine is impossible. This definition is not to be mistaken with an alternative to atheism. Atheism and theism deal with belief, agnosticism and Gnosticism deal with the basis for such belief. For example, Agnostic Atheism holds that knowledge of the divine is impossible(or currently unheld) and thus belief in God is unjustified and illogical. On the other hand, have you ever heard a theist say 'just have faith'? That is an agnostic position, as they are admitting that they have no knowledge of whether God exists, and yet still believe despite their lack of such knowlege. Gnostic atheism and theism are pretty much self-explanatory given that the definition of Gnostic to be 'believing that knowledge of the existence or nonexistence of the divine is possible or currently held'.
On the other hand, Agnosticism can be also used in a general sense, in which case it refers to not believing in something without evidence. For example, I am agnostic about the existence of invisible immaterial pink leprechauns inside my computer, causing it to work. I have no evidence that they do not exist, yet nobody would call me presumptuous for assuming they don't exist. As there is no evidence for them, the logical answer is that they don't exist. The amount of things I am agnostic about is infinite, including the infinite possibilities of extraneous things I have yet to even think of. Yet believing in such things is not illogical. Thus too, is belief in a god bounded. Unless strict and clear evidence is provided, non-belief is not illogical, in fact, it is the default logical choice
The Agnostic Theist position is quite ironic, because many religious people condemn Agnosticism, but when they say "have faith", that is a declaration of agnosticism.
Agnostic is a position that recognizes that neither the atheist nor theist sides of 'the' argument are responsible positions to take. You point this out yourself, and yet clearly don’t understand the concept you’ve presented. You said, “A belief is a conviction in some thing's truth.” and this is precisely the trouble agnostics have with atheism, it’s the unbending conviction that there is no god, without evidence to support that claim. Belief and knowledge are very different, belief lends itself to faith and opinion, but knowledge is grounded in reason, and tends to utilize things like observable and testable facts. The trouble that agnostics have with atheism and theism is that the foundation for both arguments is belief.
Theists also have difficulty wrapping their minds around the notion that an agnostic can allow for the possibility of ‘a’ god, they just don’t allow for the possibility of that theist’s god. And simply because evidence of said god hasn’t been demonstrated, doesn’t make the agnostic an atheist. (Let’s use the language we’ve agreed on.) When presented with a theist’s god, the atheist ‘believes’ that god doesn’t exist, while the agnostic wasn’t presented with quantifiable evidence showing proof of that god’s existence. The idea behind the concept is knowledge.
You’re correct, agnostic does not refer to a belief in god, even though you incorrectly keep insisting it does by calling them atheist. Gnostic is spiritual knowledge, and that doctrine of faith is what inspired the ‘term’ agnostic, but shouldn’t be confused with it’s actual meaning. The knowledge referred to in both cases are very different. What Huxley meant by the term agnostic, and what all understood agnostics mean by it is “I neither affirm nor deny…I see no reason for believing it, but, on the other hand, I have no means of disproving it.” There’s no such thing as a ‘Gnostic Atheist’ because that would require specific spiritual belief. Just as there’s no such thing as an ‘Athiest Agnostic’ because that would require the logical fallacy of trying to disprove that which hasn’t been proven. They are incompatible terms. (Again, use the language we’ve all agreed upon.)
You seem to have identified the problem with trying to disprove things that aren’t proven to exist. The burden of proof is on the one making the positive claim ie. “there is a god”, atheism is foolish, not because it requires the burden of proof to rest on the theist, but because it also makes it’s own proof-requiring claim “god does not exist”.
Hope this clears things up for you. Sorry if agnosticsm isn't as infantile or black and white as you thought. Merriam-Webster has fine site if you’re still not sure why atheist and agnostic have separate definitions.