Puddles, The first video is the procedure that I use. That is procedure that Smith recommended back in the early 70's for O/A . Somewhere along the way the oxygen valve was recommended to be shut off first with acetylene. I asked why, and the reply was everything is a fuel, even the copper tip. So if the copper tip is hot enough and you shut off the acetylene first the oxygen can cause the tip to burn and start a fire. I DO NOT AGREE with what I just repeated. My experience with O/A is if the cutting tip is too hot it will pop and go out it will not stay lit. Never had a problem with a heating/welding tip. A couple of times in a paint booth I was cutting out return saddles for the conveyor chain, the tip got so hot it would not stay lit so I cooled the tip by putting the torch head in the water of the flood sheets. All the way up to within 1/8 inch of the flame. The torch was not adjusted improperly, that conveyor is covered with paint over spray so you have the torch in a fire as you are cutting. And yes the fire does get bigger and is put out with water before it gets too big to work in.
The second video you posted is for propane. I have zero propane experience so do not know if there is a special procedure, different from O/A for light up and shut down. I did learn something from the Smith video though.
Here is a picture of one of the paint booths I have worked in. One of the sprayers in the manual section. Most of the booth is automated.
I could not get the picture to post. I need another lesson.