OK, proper oil light operation! Like RRR said, the light should ONLY come on when the key is on, motor is off (or, ugh, if you lose oil pressure for some reason).
The light should go OUT when you start the car (within 2-3 seconds) and STAY out until you shut the car off (or lose oil pressure).
When I worked on Harley's, the Twincams often had oil lights that wouldn't come on when the key was on, engine off. You started the bike, and the oil light would flash and go back out. That was a bad sending unit, and replacement always restored order.
In the OP's case, I would NOT call this fixed until you find out WHY the oil light had key-off power. Just because it stopped for now does NOT mean it is fixed (unless you did something that fixed it).