There are two meanings, one for civilian pilots and one for military.
Civilian
As an FAA controller, I was taught that "closed traffic" means "successive operations," which is in line with the Pilot/Controller Glossary:
CLOSED TRAFFIC- Successive operations involving takeoffs and landings or low approaches where the aircraft does not exit the traffic pattern.
This means the pilot will remain in the local traffic pattern (and furthermore, specifically in the left or right traffic pattern as assigned by ATC) for multiple operations (touch and goes, etc) in a row. (This includes the downwind, as you mention, but also the crosswind, base, and final of course.) The pilot does not need to ask for permission to stay in the traffic pattern each time they depart, and ATC does not need to issue specific pattern instructions each time; they only need to issue the runway clearance and any necessary spacing or sequencing control instructions.
The phraseology is "Make left/right closed traffic" or "Left/right closed traffic approved."
Military
The military apparently uses two local traffic patterns, a "closed traffic" which is kept close to the field and another one—it might be called the "open" or "outside" traffic pattern—which has much wider crosswind and base legs, so the downwind leg is flown several miles away from the runway.
Therefore in order to achieve the same effect as telling a civilian pilot "left closed traffic approved," I must tell a military pilot "continuous left closed traffic approved" or else they will "request left closed" on every single upwind. If you add that word to your phraseology you might find that they stop bothering you each time.
In your situation, you mention a pilot who said he wanted to "Stay in the pattern." To me this implies a request for closed traffic as the P/CG defines it—he will remain in the closed traffic pattern for multiple landings or low approaches. But if you are confused, you can always ask for clarification!
Other terms you might hear, especially if you are working a radar position, is "option tower" or "option radar." This means the pilot is requesting a clearance for the option and, once they depart, will either remain in the tower pattern or return for radar vectors for another instrument approach.
