Now that the term ""deep state" has entered into public usage, it is a convenient way to refer to actions and policies of the government that are taken, as the Wikipedia article says, "without regard for democratically elected leadership." Even President Trump has used the term. "Shadow government," or "cryptocracy, secret government, or invisible government," as another Wikipedia article says, carries with it more "conspiratorial" connotations -- though all of these terms include by definition the notion of conspiracy (in short, bad things planned by more than one person; see here for the legal definition).
As commentators such as Noam Chomsky have long pointed out, and as Jada Thacker has recently elaborated on, the United States has never been a "democratic" country, but a plutocracy or oligarchy from the beginning. This is nothing new from the "structural" or Marxist point of view, though those who see the failings of government as rooted in the capitalistic (and inevitably imperialistic) economic system de-emphasize the notion of "conspiracy" -- and even specifically deny it, as Chomsky does in the case of 9/11 (many links) and the assassination of JFK (also many links, but summarized here; see also my Looking for the Enemy).
As I pointed out years ago, the notion of "deep state" as somehow parallel, embedded within, or otherwise coexistent with a "public state" is an instance of doublethink, as it requires us to accept two contradictory and mutually exclusive ideas as simultaneously true. (See here and here.) In reality there can be and is only one state, one government, and that is the one that wields the power, that makes things happen, or not happen. If things happen (or not) "without regard for democratically elected leadership," it is not the "deep state" or some "shadow government" usurping power from the "public state," but simply the state being what it is and acting accordingly. It cannot be something that it is not, or something other than what its actions prove it to be. As Sartre said, "existence is prior to essence," or as we say today, "It is what it is."
What it is, as Chomsky and Thacker and many others have copiously described, is clearly not a democracy. At some point, if we are to progress, that illusion will have to abandoned. There is no "deep state" distinct from the "public state" except in this one respect: the latter is a figment of our collective historical imagination, aided and abetted 24/7 by the MSM, the educational system, and by our friends and relatives throughout the world. It is literally "the American Dream" -- the dream state.
It is tough enough to make people receptive to the kinds of fundamental change that may someday make the dream state more of a real state. Towards this end, it does not matter whether one is a "structuralist" or a "conspiracist"; whether things happen because capitalist imperialism makes it inevitable or because hidden actors make them happen, they do not happen in accordance with our dreams of democracy and "liberty and justice for all."It was in this spirit, for example, that I buried my qualms about Chomsky and others who do not share my ideas about the JFK assassination or 9/11. It is a daunting enough task to bring individual "progressives" like Chomsky and Paul Craig Roberts together, even to talk (but possible; see here and here, resulting in this), and obviously much harder to build a "movement," but there are encouraging signs (Bernie Sanders, Jill Stein). There are big differences between Roberts and Chomsky and Sanders and Stein, but they have enough in common to band together (even with some Trumpers who believed his false promises about making peace with Russia and "draining the swamp"), and part of this will be the common recognition that the "deep state" is the state, and needs radical change.