The second example uses the Waves animator. The original image has three layers. The background is just the Golden gradient that comes with GIMP. The next layer is the text "LOGIC" which has been blurred with Filters/Blur/Gaussian Blur (IIR). Incidentally, this is how I created the bump map for "metta" in the previous example. The top layer is the text "metta", with the Golden gradient applied in "Shapeburst (angular)" mode. The image was then flattened, and the Waves animator applied.
The third example uses the aptly-named Spinning Globe animator. It uses the same base image as the previous example.