One quick thing to add, once you have the properties window open, the points that define the polygon should show on top of the polygon line. If you hover over a point, it should change colors, slightly move that point a bit and release. Then draw your next point from there. This is how you start from an existing point.
Things can get screwed up here though if you proceed in the wrong direction (ie. clockwise or counter-clockwise) from the direction you were originally creating the polygon. If one direction, doesn't work, hit the delete key to delete your last erroneous points in order, and then proceed in the other direction from the next point over. If you're creating a ton of polygons, it helps to always choose a default direction of creation....but if you don't it's still easy to recognize after a couple newly created points.
Hopefully that made sense.