Most, not all, lamps that are called oil lamps use kerosene, the stuff sold as "Lamp Oil" is refined kerosene. A true oil lamp is an ancient lamp that looks like what we would recognize as Aladdin's Lamp from the fable in "Arabian Nights" many of these true oil lamps are pottery or bronze. It was common to use castor oil in these but rendered fat could be burned too though likely it had a greater significance as food.
This one I hand made and had a local ceramic shop fire it for me. It has three burners and I'm burning bacon fat. These lanters work better made out of metal when it's cold, the heat from the flame is transfered to the oil keeping it liquid and flowing. This one works fine in warm weather.
This is a very simple oil lamp I made up. I must have made 50 or 60 of these, I thought they were cool and tried to sell them for a dollar a piece I wound up selling like two of them and gave the rest away. I use canola oil or bacon fat with these. The wicks are taken from dollar store cotton deck mops.
I have an Aladdin, I've had it for almost 30 years.
I have four Dietz lanterns, I have two 18th century style copper candle lanterns that have a double round wick whale oil lamps in them, hand made by a buddy, they burn kerosene, I usually buy the ultra pure, which is just a highly refined kerosene with virtually no smell.
I have a bunch of candle lanterns too, these are a couple different style, they make candles a bit safer than just placing them in candle holders. I also use a wrought iron candle chandelier made by another friend. I make my candles with locally procured beeswax, I easily have a few hundred made up and on hand.
I have a French made Vinters Lamp with a solid round wick and a hollow round wick lamp handmade in the French Alps by the same family continuously since 1869 using tools powered by a 19th century water wheel* clipped from Lehmans Hardware.
I like primitive lighting. I've collected and restored quite a few old lamps, I have one that was used on the Orient Express back in the 19th Century.
I have a couple of the Coleman lanterns too but rarely use them, the light is harsh but they come in handy for bridge fishing.
I live in Homestead FL.