I agree that landscape photography isn't determined by the orientation of the longest image axis. Landscape has to do with the environment (the "land") being the focus of the image.
As to the two images you've posted, I'll be honest and say that they don't really catch my eye. The first image, as you've noted yourself, is completely centered. Centering isn't always a problem, and the rule of thirds doesn't apply to all images. However, in this case it is difficult to discern the point of the photo. It isn't artistic so much as documentary. You've captured the details, but seem not to have an interesting "take" on the scene. The foreground is very cluttered and distracting, as is the background. You should return to the same spot and work on isolating the cholla more, either with composition or with depth of field. As well, you can work on interesting ways of including the immediate environment in the image so that we see the relationship between the cholla and its surroundings.
The second image, while not centered in the same way, also has the problem of lacking depth. There isn't an interesting foreground element to lead the eye into the frame. In the frame itself, there isn't any center of interest, so I'm led to the conclusion that the colors are what caught your eye. Finally, the clouds are completely clipped. Clouds, especially very white clouds at sunset or sunrise, can be very difficult to capture.
What I found interesting in this image, and something definitely worth returning for, were the rounded rocks in and among the grass in the foreground. An image focusing on the grass and stones would be very compelling, IMO.
I'd like to see these new photos once you've shot and processed them. Consider these first two shots to be scouting shots. You know a good location and time of day, now go out and find the image. But, be patient as you work.