Lol I'm pretty sure the "deep state" just means, "manipulating the levers of power from a place without accountablity/oversight" which covers both these shadowy hidden layers of govt you describe and the shadowy wealthy elites funding and lobbying for whatever. It can be both.
I agree with parent, that's not how right-wing propaganda portrays the deep state. It's a strawman, of unaccountable government bureacrats. It excludes billionaires and dark money in politics (the real problem), they put themselves on the good side. (Remember DOGE?)
Sure, anything can mean whatever you want if you redefine anything to suit your whims at the time.