To note: when you write “deep state” what you really mean (as per your detailed definition) is better called “the establishment,” a term less prone to misunderstanding in wacko conspiracy theory terms.
I actually have a nuanced view of this; I don’t believe they are synonyms.
“The Establishment” is the established capitalistic order. It’s Big Business that promotes certain business-inhibited regulations and laws because they have the size and scale to manage their costs, but know that their smaller, more innovative competitors are crippled by them. It’s the financial backers of the politicians that buy support through donations (to both parties! Starbucks is just as guilty as Exxon!) and in some cases outright quid pro quo bribery. Its primary target is the politician.
The “deep state” is the established bureaucratic order that protects itself first and foremost. It transcends political parties, and focuses on the defense of political policies which keep its profitable bureaucratic structures whole and functioning. It targets threats to that bureaucratic order, whether they come from politicians like Sanders or Trump, or whistleblowers who threaten to expose the corruption within.
It’s a Venn Diagram to be sure, and we can debate the extent of the overlap. But consider that they are two different things with two different “solutions” if they are ever to be addressed.