In trying to find a meaning for atheist and agnostic, I think it will quickly become apparent that it is very important to understand the difference in meaning between believe and know. For the following, except where otherwise stated, I will take belief to mean something which a person thinks is the case, but does not claim to be one-hundred percent positive. In other words, "I believe that the sun will rise tomorrow" means that I highly expect the sun to rise tomorrow, and I would be very surprised if it didn't, but I don't discount the possibility that it won't (the laws of physics, as we know them, might change and the sun just disappear, to take one outrageous but logically possible example). I will take knowledge to mean something that the person claims is the case for sure, and is one-hundred percent positive there is no possibility it is not the case. In other words, "I know that the sun will rise tomorrow" means that I not only expect the sun to rise, but I don't consider it possible that the sun won't rise - it just can't happen. Thus the difference is one of probability, of how sure a person is about what they think/claim. I also take belief and knowledge to be exclusive (i.e. knowledge would be considering the probability one, and belief would be considering the probability greater than zero but less than one, as opposed to greater than zero but less than or equal to one, which would trivially make all knowledge belief as well).

The following are some common meanings of atheist:
(AT1) A person who believes that no gods exist.
(AT2) A person who knows that no gods exist.
(AT3) A person who believes that a specific god (or number of specific gods) does not exist.
(AT4) A person who knows that a specific god (or number of specific gods) does not exist.
(AT5) A person who lacks a belief in regards to the existence of gods.
(AT6) A person who lacks a belief in regards to the existence of a specific god (or number of specific gods).

The following are some common meanings of agnostic:
(AG1) A person who lacks knowledge in regards to the existence of gods.
(AG2) A person who lacks knowledge in regards to the existence of a specific god (or number of specific gods).
(AG3) A person who lacks knowledge in regards to the existence of gods, but knows such knowledge is possible.
(AG4) A person who lacks knowledge in regards to the existence of gods, but believes such knowledge is possible.
(AG5) A person who believes that knowledge in regards to the existence of gods is impossible.
(AG6) A person who knows that knowledge in regards to the existence of gods is impossible.

The distinction between the belief ones and the knowledge ones above, as with (AT1) and (AT2) or (AG5) and (AG6) should be clear given the meanings of belief and knowledge I put forth at the beginning. It should also be noted that atheism seems to deal with the existence of gods, while agnosticism seems to deal with knowledge (or more specifically, the lack thereof) of the existence of gods.

Now, (AT3) and (AT4) aren't really used all that commonly as definitions for the word atheist, but I included them because they are common as claims that people make because they are compatible with some of the other definitions given. For example, a person can be (AT1) and (AT4), in that they leave open the possibility that some god or gods might exist, but leave open no possibility that certain gods (like God from the Bible) exist, perhaps because the god/s is/are contradictory. Thus, the person might call themselves an atheist meaning (AT1), but still claim (AT4) when presented with notions like God from the Bible.

I should clarify (AT5) and (AG1). By (AT5), I mean someone who doesn't believe that no gods exist and who doesn't believe that one or more gods exist. By (AG1), I mean someone who doesn't know that no gods exist and who doesn't know that one or more gods exist.

It is interesting to note the many different possibilities (including combinations of the definitions given above) of what a person could claim where they would still likely call themselves an atheist and/or agnostic. Examples of compatible combinations:

(AT1) is compatible with (AT3), (AT4) and (AG1-5).
(AT3) is compatible with (AT5), (AG1-5) and even with (AT4) and (AT6) given that the specific god(s) in them are not the same as the specific god(s) in (AT3).
(AT5) is compatible with (AT2), (AT3), (AT4), (AT6) and (AG1-5).
(AG1) is compatible with (AT1), (AT3-6) and (AG2-6).
(AG2) is compatible with (AT1), (AT3), (AT5), (AT6), (AG1), (AG3-6) and even (AT4) given that the specific god(s) in it are not the same as the specific god(s) in (AG2).

It should be obvious by now just how many possible meanings and claims are covered under the two words atheist and agnostic, and it should also be obvious why so many people in the atheist/agnostic community end up arguing over meaning. What's the solution then? We could define atheist as one of (AT1-6), or one of (AT1-6) and some or all of those it is compatible with (as long as no two of those it is compatible with are incompatible). Similarly, we could define agnostic as one of (AG1-6), or one of (AG1-6) and some or all of those it is compatible with (as long as no two of those it is compatible with are incompatible). Then we would have to invent new words to cover the other possibilities. Some people have done just this by creating the weak/strong atheist dichotomy, where a weak atheist claims (AT1) and a strong atheist claims (AT2). Obviously though, to cover all the possible claims one could make regarding the existence of gods and knowledge of the existence of gods, quite a few new words would need to be created, and that really doesn't seem very pragmatic.

I think the best solution is simply to realize that language is often vague and fluid, that it's incomplete as a formal system, and that we shouldn't expect to have a "perfect" definition for everything. In the end, I think the most important thing is not that everyone agrees on an exact definition of the words so that their use is always completely non-vague (if that's even possible), but that people understand the ideas behind the words - that they understand what a person means when they use those words. Thus, it becomes a matter of context - in some cases, it is enough for a person to describe themselves as an atheist and get their meaning across; in other cases, it is prudent to elaborate and clarify.

Note: A problem that I haven't even approached here is what is meant by 'god', which would appear to be quite relevant to the discussion at hand. Some people take 'god' to mean a specific deity from a specific religion (example: the Judeo-Christian God or the Muslim Allah would thus be considered gods). Other people consider any conscious being with unlimited powers (or alternately, powers limited only such that a contradiction cannot happen), and/or unlimited knowledge to be a 'god'. Still others, who consider a first cause necessary for existence, equate that first cause with 'god' (or perhaps only if that first cause is conscious). Some even use the word 'god' as a synonym for the universe, for "all that is" (Spinoza and Einstein would likely fall into this category). 'God' seems to have as many definitions as atheist and agnostic, if not more. I won't attempt to set forth a certain specific idea of 'god' here because the many meanings it carries simply add more variables to the possibilities of what is meant by atheism or agnosticism. As before, there are times when it is prudent to elaborate and clarify what is meant by 'god' and times when it is not necessary.