My perception is that distinctions of the sort you describe can be found but that they are both largley modern and contextual. So, one might determine the distinction in Hegel, Rawls, Foucault, etc. rather than find a uniform distinction across texts. A quick search of JSTOR raises this article that seems to offer some historical contextualization:
“Theories of the Origin of the State in Classical Political Philosophy” by Harry Elmer Barnes, in a journal called “The Monist.”
Vol. 34, No. 1 (January, 1924), pp. 15-62.
Dwight Waldo’s book, The Administrative State (1948 but reissued in 2017 by Routledge), is a classic and makes an interesting distinction between the administrative and welfare state that may be helpful to you.
As for the importance of the distinction, I leave that to others with more expertise in political philosophy, but my perception is that it is not terribly central. You will find some discussion of elite theory among political scientists. Trotsky’s critique of the Stalinist USSR comes to mind as relatively important, too.
Less relevant to your main question is your elaboration of the distinction. That elaboration seems tendentiously freighted—and I’m curious about whether there’s a connection between the distinction you draw and the politics of your elaboration. You do raise important philosophical questions about the proper role and function of the state and what sort of compensation and taxation policies are fair and just. What strikes me as tendentious is that you use the practices of private economic organizations as a standard against which to measure those of the government. Why not the other way around? Why not use the public to judge the private? Perhaps it’s not that workers and officials in government are treating themselves and others improperly but that the private sector is treating owners and employees improperly. Perhaps the unfair, self-interested conduct is not properly located among government workers but among the owners and managerial class of the private sector who have hoarded for themselves the wealth generated by the economy, leaving others unjustly without pensions and generally with diminished compensation.
I lived during a time when private sector pensions and medical benefits were much more extensive and substantial than they are today, and I saw them whittled away over the decades. Perhaps that was the injustice. Similarly, perhaps the trouble is not that government employees wish to hold onto their benefits in the face of deficits but that deficits have been unjustly created either deliberately, through incompetence, or though neglect by those responsible for securing state revenue through taxation, etc. Perhaps through the use of their assets in political donations, lobbying, think tanks, etc., the wealthy have improperly advanced an ideology of austerity at the expense of the polity generally.
—Of course, this alternative view is arguably tendentious, too. I raise it to illuminate the apparent implication of the way you put things. You may have thought this all through on a philosophical level, but it may instead be that the distinction you’re after is a tool of political activism and ideology of which you’re unaware rather than sober political philosophy.