The words "anarchy" and "anarchism" can´t be reduced to being against the state and to "anti-statism". There are plenty of non state and even anti-state social groups that are hierarchical and authoritarian such as capitalist businesses and mafia and gang organizations and so they cannot be considered "anarchy" in existence or "anarchist" groups because of their being hierarchical (they have hierarchies, ranks, bosses and in the case of bussinesses specific "owners")
Anarchism means being against hierarchies and not just those present or created by the state and so perhaps the closest word to anarchism might be "anti-authoritarianism". Non-state hierarchies that have been fought by anarchists include class hierarchy at the workplace and at production, those created by religion and church, educational in school contexts both state, religious and private, and also others such as patriarchy.
Democracy can happen both at a state and outside it. For example in an assembly of a small collective, if things are decided on consensus, that is a democratic form of reaching a decision. The US Congress also reaches decisions through a democratic process. It is clear though that the difference there is the existence of a state, representation through elected congressmen, the existence of a central federal government and the power of money on campaigning. As such one should make a distinction between democracy as mediated by state and capital and democracy not mediated by those things.