I always was under the assumption, that the FBI is a police-force, not a spy-agency (like the CIA or NSA). I kind of realize that the difference is getting smaller in recent years.
There is good reason to keep them (Police and Intelligence) separated, in germany we call this 'Trennungsgebot', this is a direct historical consequence of the Gestapo.
So, technically, is the FBI a police or an intelligence agency?
The FBI is an investigative law enforcement agency. So, I would say that they are intended to be both.
The CIA, a "pure" intelligence agency, has been competing with the FBI for decades, leading to the FBI becoming more and more of an intelligence agency over time (in addition to their law enforcement "police" activities).
