That light should turn due to low oil pressure. Typically that happens due to low oil, which you already verified.
There are some other reasons for that light to turn on.
1. Low oil pressure due to faulty oil pump, clogged passages, bearing clearance.
2. Bad oil pressure sending unit (on back of the engine).
3. The wire from the oil pressure sending unit to the instrument cluster is grounding.
4. The integrated control module is shorting (that is wired to the pressure switch too).
5. A problem with the instrument cluster main board.
#1 would be the most concerning to me, because that would mean that you oil pressure is actually low. The only way to verify is to remove the pressure sending unit and connect an oil pressure gauge.
The one test I would try is to unplug the one wire connector to the oil pressure sending unit. Now the light should not turn on at all when the engine is running. From the wiring, I don't know if it will do the bulb check, let us know if it does. If the light does flicker, then you can rule out low oil pressure and the sending unit.
That sending unit is a bit above the oil filter on the back of the engine block. It is right above the metal pipe that goes from the water pump to the thermostat. It will have one wire (yel/red) going to it and a rubber boot that covers the sensor. That boot will seem like it is plastic after 15 years of engine heat.
If the light doesn't turn on with it unplugged, then I would replace the oil sending unit. They cost <$20 at the dealership. They use a special oil sending unit socket to install, so you don't damage the sending unit. You are also supposed to use a light coat of RTV on the threads. The torque is 13 ft-lb.
2005 Acura TSX MT