I think it's you who are misunderstanding. The definition of "spy" that I know is that they are people who cultivate relationships with "agents" or "assets": people who have access to important secret information and are willing to share it.

Most spies don't go into forbidden places and physically dig up secrets themselves. It's infinitely easier to get to know someone who already knows those secrets and persuade them to tell.

That's not the correct definition. A spy is someone who does spying, and spying is obtaining information illicitly. Like the Cambridge Five:

https://en.wikipedia.org/wiki/Cambridge_Five

Or Aldrich Ames:

https://www.fbi.gov/history/famous-cases/aldrich-ames

Someone who manages spies is an intelligence officer.

I think that just depends on your definition of spy. If you're thinking, say, Americans being undercover or sneaking onto bases (instead of using local assets to do it) then it would probably fall under the Special Activities Center, which is a subset of the CIA.

While you're right that these folks aren't undercover in the sense that their identities are known, that those identities are linked to an intelligence gathering agency is not known. A lot does happen out in the open and it always has. Being a member of the diplomatic service gives them protection when they're operating in a foreign country. You operate out of an embassy and if you're arrested they usually trade you back to your home country, because diplomatic immunity.