Three-jaw chucks are usually a little inaccurate, holding the stock slightly off the lathe's spindle axis. So an initial light cut will only shave one side of the work. However, as you go in deeper so that the tool is cutting all round, the result will be circular in cross-section, and concentric with the spindle axis. That is fine if you are making a part which can be finished without reversing it in the chuck, and will simply be cut off from the parent stock once it is finished. If you do reverse it as you have done in your video and then try (for example) to turn the other end to the same diameter, the two ends won't match up.
It may well be that you will end up buying a 4-jaw chuck with jaws which can be independently adjusted to get the work dead on centre, with the aid of a dial indicator or "clock" bearing on one side of the work. Another technique is to turn the work "between centres", so you can turn it along its whole length.
Andy