J. Edgar Hoover was a perfect example of what people are talking about with the deep state. You can claim that’s just “institutional inertia”, but it’s still some unelected official with massive power wielding it in nefarious ways.
It's a very imprecise term covering bureaucratic norms and blindspots, to unsanctioned domestic and international terrorism, so it's not really useful except as shorthand with ideological bedfellows, so-to-speak, and you're right that it's often used to mean "the government guys I don't like but am too lazy/ill-informed to describe". But OP mentioned Gladio -- I think that's context enough for this thread. Do some reading on the topic (beyond Wikipedia) if you haven't already, and you might get a sense of what this non-centralized deep state can do.
And I didn't make it clear, but I brought up the North et al "shadow government" as an example far on the small-scale end of the spectrum of meaning of "deep state/shadow government". It _was_ a centralized shadowy cabal doing a now-documented conspiracy. People argue about how much authority they "seized" and how much was delegated by Reagan (and how much capacity he had to delegate much of anything), but regardless, it was not democratic to illegal go around Congress. What I've described as the "deep state" is the more common usage, and it's on the "amorphous, non-centralized, competing institutions, largely free of any democratic accountability" end.
"The Government" isn't really a single institution, but a surprisingly loose collection of various institutions that all have some level of momentum guiding the direction they move in. It's critical to note that these institutions may have interests that harshly conflict with one another. Even the intelligence community, insular as it is, cannot be thought of as a monolith, but as various institutions with various priorities, some which conflict with the interests of other government institutions.
When the government does something we dislike politically, or does something we don't understand, it's "the deep state" or "the shadow government" which betrays a deep misunderstanding of what the government is. There's no central, shadowy cabal controlling the affairs of "the government" from behind the scenes, the various institutions that comprise the government maintain momentum and do what is in furtherance of their particular interests.