Alexandra-san, Thank you for sharing the site. The instructor explains the terms pretty well! I just would like to add that うまい tends to be used in a casual conversation.
You should also check this link about the difference between 上手・とくい and 下手・にがて: http://www.rocketlanguages.com/your-community/japanese-vocab/good-or-bad-at/
Long story short: 上手・下手 when you are talking about somebody else and とくい・にがて when you are talking about yourself. :)