during my initial training, I took at 172 out for a spin to practice slow flight and steep turns. On my way back, the oil door on the nose popped open. There was no visible leaking and oil pressure remained constant and in the green, so I continued without even notifying ATC, landed normally and taxied to the flight school. Turned out the latching mechanism was a little worn, they had it replaced the next day.
Of the recordings I've heard of GA pilots getting into situations one might consider an "emergency," more often then not the pilot describes a problem but never outright says "I declare an emergency." Understandable, especially when you don't want to admit there's a problem, or you think about the discussion we're having right now about what does and does not qualify. For example, a 172 pilot who notices his flaps aren't coming out might inform ATC of the problem, but might not. Flown right, and with enough runway, a 172 doesn't need the flaps to land. If the field is short, or traffic high, ATC has a better chance of being told. Even then, I'm not sure I'd call it an "emergency" so much as a "heads up."
As far as your scenario goes, there are many other factors involved. The pilots would run through the checklist for low oil pressure indication, and try and troubleshoot the issue. How long has the light been on? are there other signs of an issue? (low engine RPM, low fuel flow, low oil indicator, etc). If they were a little low on fuel, they'd have to decide if they thought traffic would let them get in ok, if they needed to divert to an alternate, or if they needed to declare a fuel emergency. Do you have enough fuel to divert? Do you have enough to attempt a landing? Is weather bad? If it is bad, do you have enough to attempt 2 or more landings? Can you attempt a landing at the bad weather airport, but have enough fuel to go to the alternate?
The most likely scenario for low fuel in the airline department is a diversion. If JFK is slammed, the weather is deteriorating, and you don't think you have enough fuel to sit in a holding patter for an hour and a half, you're going to go someplace else.