Actually it's two totally different processes.
Carbon arc gouging, air arcing, gouging uses a ....surprise LOL...carbon rod 1/4" + or so in diameter. You need a welder with lots of amps and a compressor with lots of air. You hold an arc between the item you're gouging and the carbon rod which turns the metal molten. The air is directed in the area of the arc and blows the molten metal away. Big shower of sparks which travel a significant distance. More commonly used for backgouging welds or removing broken/cracked welds to get to good material.
A thermal lance, slice torch, is an exothermic process, once you get metal rod hot enough, you feed pure oxygen to it and it will be self sustaining i.e. burning. That's why people using torches in a shipyard for example and called burners not cutters. You can do the same thing with an acetylene torch if you have a piece of clean plate. Start the cut and turn off the acetylene and oxygen, just using the cutting oxygen and you can keep the cut going......Mike