That's right, the best is to have a watered dishcloth that will suffocate the fire, hence stopping it. The fire extinguisher may be a bad idea because the pressure will spread burning oil around across the kitchen and the water is definitively the worse idea because it will counterintuitively make a bigger flame and also spread burning oil.

Watered dishcloth? Water is the absolute last thing you want near very hot oil, never mind burning oil.

You're right, the best is a metal lid.

That works, but be sure to use a wool or cotton cloth.

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The proper method to deal with an oil fire: turn off the heat and smother the fire with a metal lid, metal baking sheet, or wool/cotton cloth. Do not use a fire blanket made of polyester or nylon; it will melt and burn producing toxic smoke. Do not use flour or any other flammable/explosive powder. Do not use water, as the liquid water will get into the oil and expand 1600 times its original size as it turns to steam; this rapid expansion will fling burning oil everywhere. If you do use an extinguisher, be careful not to spread the burning oil.

For small NON-OIL, non-electrical fires, smothering with a wet non-synthetic cloth is the best way to stop the fire.

When it happened here, I instinctively threw the flaming pan out the adjacent open door into the yard. We ate something else that day.