There is much confusion around the term anarchy and it's political ideological counterpart. We all understand the customary use of anarchy to mean "complete disorder and chaos" but the etymology of the word breaks it down. From medieval Latin from Greek anarkhia, from anarkhos, from an-‘without’ + arkhos ‘chief, ruler.’ Anarchy is the "absence of government and absolute freedom of the individual, regarded as a political ideal", by definition. Which I refer to as "Anarchy", always capitalized when it necessarily shouldn't be grammatically. I'm an anarchist however and believe in doing things how I see fit, not by the "rules". Did you see what I just did there? The lowercase form of "anarchy" to me is always, "a state of disorder due to absence or nonrecognition of authority" yet again by definition. The delineation of the two is paramount.
Anarchism and anarchism
Proudhon stated that Anarchy is "the absence of a master, of a sovereign." Now, Anarchism by definition is, "belief in the abolition of all government and the organization of society on a voluntary, cooperative basis without recourse to force or compulsion." If you agree with that then you believe in Anarchism. The lowercase form of anarchism is, "the advocacy or practice of anarchistic beliefs." Which anyone that is anarchistic in nature is, "a person who rebels against any authority, established order, or ruling power." Therefore a person who is an anarchist could still believe in Statism and not actually Anarchism at all. They wouldn't be considered an Anarchist. This is how I've differentiated the two, think of it as using "than" or "then".
So what does it ultimately mean to be an Anarchist or an anarchist? That question is solely left up for you to answer. I merely thought I could help in separating the two for you by the use of a capitalized and lowercase form.
The Anarchist in me wishes to end this by quoting Molinari, "Anarchy is no guarantee that some people won't kill, injure, kidnap, defraud, or steal from others. Government is a guarantee that some will." The anarchist on the other hand will leave you this quote from Lorca, "The artist, and particularly the poet, is always an anarchist in the best sense of the word. He must heed only the call that arises within him from three strong voices: the voice of death, with all its foreboding, the voice of love and the voice of art."