Let’s start with the basics. Atheism is just thinking there aren’t any gods. (Theism, similarly, means thinking that there is a god or gods.) Some people will define atheism as being even less than that, that atheism is mere lack of a belief in God. I don’t bother with that definition, because I’m happy to say I think there aren’t any gods. If a child, say one who’d just watched Disney’s Hercules, asked me if the Greek gods are real, I’d say “no” rather than hedge my bets by saying I lack a belief in them. And I don’t see any reason to treat other gods any differently.
But I don’t think the difference between those two definitions of atheism is very important. What is important is that when we talk about atheism, we use a definition of atheism that’s at least close to definitions used by actual atheists. That’s important because using definitions unlike anything any atheist actually uses creates needless confusion. It also gives believers a way to pretend to refute atheism (in the sense of what actual atheists actually believe) without actually saying anything about atheism.
Here’s how that works: a believer starts talking about atheism, but then switches to talking about the evils of “naturalism.” Or about the evils of “philosophical materialism.” Or “scientism,” or “nihilism,” or any other philosophy known for scaring small children. For example, theologian John Haught, in a book written to respond to Dawkins, Harris, and the rest, tells his readers that if they are atheists, “you will be required by the logic of any consistent skepticism to pass through the disorienting wilderness of nihilism… Are you willing to risk madness? If not, then you are not really an atheist.” 
This is nonsense. Plenty of people see no reason to pass through any metaphorical wilderness, or risk any sort of madness, but are still real atheists because they think there aren’t any gods. What definition of “atheism” is Haught even using? Not one used by any actual atheist, as far as I can tell. And for all his talk of logic, he never explains why he thinks what he says is true.
Some atheist philosophers do also call themselves “naturalists” (sometimes, “philosophical naturalists” or “scientific naturalists”). Talk about “naturalism” can refer to many different debates in philosophy, and I don’t care about most of them, but it does of course make sense other philosophers who care about those debates to sometimes criticize these naturalists. However, disproving naturalism (or any other philosophical view) still wouldn’t show that there are any gods. 
Because atheism is just thinking there aren’t any gods, if you think there aren’t any gods, you’re an atheist. But many atheists refuse to call themselves atheists, probably because they don’t want people taking offense at their mere existence, demanding they risk madness, and so on. In the 2008 American Religious Identification Survey, 2.3% of people said “there is no such thing” as God, but only 0.7% called themselves atheists.
The situation was even worse with agnostics (“agnostic” means someone who says they don’t know if God exists.) 10% of people said “There is no way to know” or “I’m not sure” in response to the God question, but only 0.9% identified as agnostics. This is surprising, because the word “agnostic” is much less stigmatized than “atheist.” 
I sympathize with atheists who don’t want to call themselves atheists because they don’t want trouble from believers. But I’m still going to say to them: if you’re an atheist (or an agnostic), say so! The way atheists and agnostics get accepted by the general population is to be visible and get believers used to the idea that there are lots of atheists and agnostics in the country. That requires telling people we’re atheists and agnostics.
By the way, while atheists are often critical religion, atheism isn’t the same thing as being critical of religion. You can be an atheist without being exactly like Richard Dawkins. Personally, I like the man, but the world doesn’t need a million Dawkins clones. The world has plenty of problems worth caring about, and not everybody needs to care about the same ones. And as a matter of fact, while many atheists are very critical of religion, others are apathetic, and some have even argued we shouldn’t criticize religion, at least not too loudly. (Obviously, I disagree with this last group of atheists.)
There have also been people like Thomas Paine (1737-1809), the American revolutionary writer who believed in God but whose book The Age of Reason was harshly critical of Christianity. If you enjoy reading Dawkins, then most likely you’d enjoy reading Paine more than you’d enjoy reading the atheists who think critics of religion should be quiet.