There is, unfortunately, some disagreement about the definition of atheism. It is interesting to note that most of that disagreement comes from theists — atheists themselves tend to agree on what atheism means. Christians in particular dispute the definition used by atheists and insist that atheism means something very different.
The broader, and more common, understanding of atheism among atheists is quite simply "not believing in any gods." No claims or denials are made — an atheist is just a person who does not happen to be a theist. Sometimes this broader understanding is called "weak" or "implicit" atheism. Most good, complete dictionaries readily support this.
There also exists a narrower sort of atheism, sometimes called "strong" or "explicit" atheism. With this type, the atheist explicitly denies the existence of any gods — making a strong claim which will deserve support at some point. Some atheists do this and others may do this with regards to certain specific gods but not with others. Thus, a person may lack belief in one god, but deny the existence of another god.
Unfortunately, misunderstandings arise because many theists imagine that all atheists fit this most narrow, limited form of the concept of atheism. Reliance upon dishonest apologists and cheap dictionaries only exacerbates the problem. So, when someone identifies themselves as an atheist, all you can do is assume that they lack belief in the existence of any gods. You cannot assume that they deny any gods or some particular god — if you want to find out about that, you will have to ask.
Why do these errors occur? Why do some theists insist that the broader sense of atheism simply does not exist? Possibly some theists feel that since they are claiming the existence of their god, then anyone who does not agree with them must be claiming the exact opposite — a serious misunderstanding of not only basic logic but also how human belief systems operate.
Another reason for insisting that only the narrow sense of atheism is relevant is that it allows the theist to avoid shouldering the principal burden of proof. You see, if atheism is simply the absence of a belief in any gods, then the principal burden of proof lies solely with the theist. If the theist cannot demonstrate that their belief is reasonable and justified, then atheism is automatically credible and rational. When a person is unable to do this, it can be easier to claim that others are in the same boat than to admit one's own failure.
There is also a tendency among some theists to make the error of focusing only on the specific god in which they believe, failing to recognize the fact that atheists don't focus on that god. Atheism has to involve all gods, not simply one god — and an atheist can often approach different gods in different ways, depending upon what is necessitated by the nature of the god in question.
Thus, when someone claims that a person is an atheist because they "deny the existence of God," we can start to see some of the errors and misunderstandings that statement involves. First, the term "God" hasn't been defined — so what the atheist thinks of it cannot be automatically assumed. The theist cannot simply assert that whatever they have in mind must also be something which the atheist has in mind. Second, it is not true that whatever this god turns out to be, the atheist must automatically deny it. This concept might turn out to be too incoherent to justify either belief or denial.
As a matter of fact, many exchanges between atheists and theists turn out to be frustrating and unsatisfactory because no one ever bothers to stop and explain what is meant by the key term "god." Unless and until that happens, no serious, productive, or rational discussion can take place. Unless we know what the theist means by "god," we'll never have any chance to judge if anything said in defense of belief is adequate. Only when we know what the theist means by "god," will we be able to seriously critique their concepts.