Brian,
I have found that any of the ways you mention work fine. The key is to avoid enlarging the plug hole if possible. Another technique that often works very well is to drill a hole in the center of the old plug just large enough for your screwdriver. Then begin backing out the screw. As you do that, the screw will often lift out the remainder of the plug as it comes out. If you do accidentally enlarge the plug hole, don’t despair. Simply drill a larger hole and use a larger plug.
As for replugging...although you can buy a bag of teak plugs, I often prefer to make my own. I do this especially if I am replacing wood so the plugs come from the same board and match well. You can buy plug cutting bits for a drill that make tapered plugs that will fill a slightly enlarged old plug hole. You can buy the next size up if you might need one. The tapered plugs hammer in tightly and are best with a bit of glue of your choice. My current choice is Titebond III, though epoxy or other glues are fine. With a tight fitting tapered plug often no glue is needed and varnish on top will help to secure it. There is no right or wrong way. Try to align the grain of the plug with the wood substrate as you insert it. A sharp chisel or a flexible Japanese pull saw will cut it off. If you cut it a bit proud, sand paper will finish it perfectly. The only annoying thing about this process is that you need to apply more varnish with the wood replaced so the plugs match.
Good luck!
Mike Jacker