I generally use "anarchist" to describe my political philosophy. I'm pretty sure I'm using it correctly, but I'm not certain. I haven't had much contact with other "anarchists", just a bit of exposure through history and such.
First off, to me, "anarchism" doesn't mean "no government". Rather it means "no intrinsic authority". What I see among historical anarchists is an opposition to practices that, frankly, aren't all that often practiced any more, in the political realm. I'm referring to rule by bloodline and such, nobility and royalty. I get the impression the early anarchists wanted to do away with royal governance, in favor of a federation of voluntary governments instituted at the local level. Which is to say, they believed in government; they just wanted to do away with imposed external authority.
But I do see our current economic relations as having a great deal of externally imposed authority in it... though going into my beliefs about why, and what could be done about it, would be beyond the scope of this essay.
To me, anarchism means the following:
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Favoring no unnecessary relationships of authority.
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Where authority is necessary, it should be granted by those over whom the authority is exercised, directly and individually, to the greatest extent practicable. So, for example, if we have an economic system that leaves both employers and employees with the same level of market power (we do not, but if we did), the employer-employee relationship would qualify, since it commences by choice of both parties, and can end by the choice of either party.
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Where this is impracticable, the authority in question should always be temporary, with a clearly delineated end. For example, the parent-child relationship is necessarily one of authority, since children lack the faculties to make all the decisions one needs to make. But this relationship should be premised on preparing the child to survive outside this relationship, and have a clear end point (the point of their majority). And I mainly include this but just for the parent-child relationship; I can't think of any others.
All this being said, I know there are those for whom Anarchism means "no government", usually detractors who don't actually understand the philosophy... or so I assume. Do I assume incorrectly? Is my use of the term wildly incorrect? I really don't know.
To start things off, I identify myself as an anarchist, anarcho-daoist to be exact. But I do believe, that it's a philosophical branch so varied and personal every single one of us is free to hand-pick certain memes and apply it to one's life. So to me, everyone has their own "anarchism". They are all perfectly valid to me, as long as they are based on individualism and the lack of authority. And I think that's the key. Being your own person and setting your own rules, bars and things to follow. But also respecting the very individualism of the others.