Your two photos suggest to me that you may need a little help with that "rule." Its primary purpose is to help with the placement of your subject/main point of interest, which should fall somewhere near one of the four 'power points' in the image. These power points lie where the divisions of the thirds intersect:
I think a better name for the rule of thirds would be "The Guideline Against the Center and the Edges," but that's not nearly so catchy as a title.
The basic idea is that the yellow zones (in this graphic) should be favored over the grey ones when it comes to placing your subject or secondary points of interest in the frame.
The edges are weak areas of the picture, and the center is so strong that the eye needs some compelling incentive to leave it and move around the frame. If you put your subject near an edge, it loses visual weight; if you put it in the center, it tends to create a static image. Placing your subject in the yellow zones usually works out the best. You'll know when you can break these 'rules' by understanding why they work. If something about the picture offsets the negatives of the grey zones, then you can get away with breaking the rules.
The horizontal and vertical lines in the graphic are where strong horizontals and verticals ought to fall (you did use one of them where your hills meet the sky). The reason for this 'rule' is to prevent the image from 'breaking apart' at the center. Strong lines that fall near the center have a tendency to bisect an image, so they will be a problem unless something in the picture ties the two bisected areas together quite strongly.
Hope that helps.
