Good point, was it understood in the US as anarchism at the time, though?

Communism is an anarchist philosophy, at it's root. If it seems a bit weird because (IIRC) this historical dialectic goes:

- bourgeois revolution

- Workers revolution

- Bigger state

- Even bigger state

- Underpants gnomes

- Anarchist utopia

Only half joking... this is what turn of the last century communists believed.

As the saying goes, if everyone is part of the state, nobody is part of the state. At some scale, human organizations start to break down.

anarchists (well maybe except for insurrectionist anarchists) believe in communism as an outcome, but they most definitely do not agree with vanguardism as a means of achieving it. vanguardism is just a kind of elitist authoritarianism.

I would phrase it this way: (non-communist) anarchists believe in communal living as an outcome

Aware that this isn't historically accurate terminology but very few people will appreciate the nuances your definitions requires, as the modern meanings have diverged