The world is something we are in, and is external to us.
The world is something existing in some way independently of minds.
The world is something continuing to exist when we are not perceiving it.
Externality is a phenomenal characteristic -- it's a way the world seems, and as is often the case with appearances we can hold that it seems that way either because of the way the world is or because of the way our minds make it. Call the former received externality and the latter constructed externality.
The externality of the world can be taken as a given, on either explanation of it, but independence and continuance are much harder to ground, and problems have occasionally been raised for each by this philosophical position or that. Given this, there are eight kinds of ways we could take the external world to be.
EW1, overwhelmingly the most common position, is external world realism in a strict and proper sense: the world continues to exist as an entity distinct from and independent of any mind, and it seems like it is external because it is. As far as I know, nobody has ever proposed an EW2 account, in which the world is really external to us and independent of us but only exists when we perceive it, for the obvious reason that there would need to be some explanation for how the world always ends up existing when we perceive it, and only when we perceive it if its existence does not depend on us. An EW3 or EW4 account would be only marginally less puzzling, because of the difficulty of explaining how externality is not mentally constructed if the external world's existence is dependent on a mind.
Berkeley's idealism is an EW7 account: our sense of the externality of the world is built out of our understanding of the 'grammar' of how our ideas are associated with each other, the world (since it consists only of ideas) is entirely dependent on minds, but it continues to exist in God's mind when we are not perceiving it. An EW8 account seems to require solipsism, or something practically distinguishable from it. Transcendental idealism, or at least some version of it, would seem to be an EW5 account. EW6 seems to run into problems analogous to those we find in an EW2 account.
But there is actually very little work done on the general shape of the external world problem as a whole, or about the internal dynamics of possible accounts of the external world.