This is not as complicated as it looks. A lot of the information is for little problems that can arise, but generally do not, so you will skip those parts.
The most difficult part of installing a flute stopper plug can be removing the old plug assembly. Partially unscrew the crown so there is some space between the crown and the head joint tube. Place the crown on a padded surface to protect it (a padded chair or a carpeted rug are good for this). Apply increasing pressure. DO NOT APPLY EXTREME PRESSURE OR JUMP UP AND DOWN. Some firm pressure should move the cork assembly toward the mouthpiece. If there is movement, unscrew the crown almost all the way and apply pressure again. Now unscrew and remove the crown. You should be able to push the cork out the rest of the way now that it is started. Something as simple as the cleaning rod should do. Only push toward the mouthpiece end of the head joint, as it gets wider in that direction.
If you are not able to remove the cork assembly, don't feel bad. They can get very stuck. At this point a professional with a proper tool for removing the cork assembly should be contacted.
If you removed the assembly continue: The top metal mounting plate (towards the crown of the flute) should easily unscrew in a counter clockwise direction. There may be a separate little nut holding it in place, or the plate itself may be threaded. A small pliers should do the job. Use some fabric in the jaws of the pliers here (and anytime you need to hold a metal part) to protect the metal from scratches & nicks. Now unscrew the cork in a counter clockwise direction. If it starts to break apart, use a utility knife or razor blade to break the seal with the bottom plate. Just ease it in from the side & cut around. TA----DA ! Clean the bottom plate and the threaded rod to clear cork debris and old sealant. Fine steel wool will work for this. Put some sealant (cork cement or caulk will work) thinly on the top and bottom bottom plate surfaces that contact the stopper plug. Slide (or screw if the fit is tight) the stopper plug onto the threaded rod with the side labelled "TOP" facing the crown. The threaded rod will likely be a loose fit, and this is ok.
Screw the top plate on so it is firmly in place, making sure the plug is centered on the plates. Wipe off any excess sealant that squeezed out. (A little excess all around indicates a good seal) Almost There ! Use a little lubricant (cork grease or vegetable oil work well) on the 3 'O' rings. Drop the assembly into the head joint from the wider end with the threaded rod toward the crown end. You should see the end of the plug through the mouthpiece opening, but not quite into the opening. You will need your cleaning rod to slowly push the Stopper Plug until the crown can be threaded on. The stopper plug should be a nice tight fit. I will center itself as it is pushed into place. Use the crown to adjust the final position using the cleaning rod tuning mark to position the stopper plug. PLAY YOUR FLUTE !!!