Yeah, you got it.
No, the feature is worded correctly. The only problem is that the devs did not mark the appropriate moves on the Pokedex like they should have. You shouldn't have to use another website to play the game.
One thing that is overlooked is that you can move some of your lower level, but good moves onto your Technical move list rather then your Natural. This is a good way to get around the 7 Natural move limit. Not useful at this level, of course, but you can do this in later levels to great effect if you are having difficulty deciding what moves to dump.