There are sub-families of anarchism, and you would be correct that the predominant form at the moment is a flavor of socialist anarchy. The purported relation to anarchy is that the world would be split into tons of small, self-organized communities that individuals are absolutely free to join and leave at will.

I tend to agree that it makes far more sense to call it socialism with some individualist facets than anarchy with some socialist attributes.

What you’re describing would be closer to individualist anarchy or philosophical anarchy. Individualist anarchy believes the right of the individual is paramount, excepting when the rights of two individuals clash. Philosophical anarchy is the general belief that the desires of individuals should not never be co-opted because one person can never morally justify forcing another to do something and thus governments can never be moral as their entire reason to exist is to wield the monopoly on violence against individuals to override their will. Individuals are of course still free to join groups and abide their rules if they choose, but those groups would not be able to enforce any kind of agenda against its members.