When I use the crop tool to crop an image and then export the result, the output is uncropped, i.e., has the original size and includes content outside the cropping rectangle. For all intents and puposes, the image appears cropped in Gimp. In fact, the pixel size displayed in the title bar is the cropped size and that is also shows under image - change canvas size. In older versions of gimp, cropping an image simply, well, cropped the image ...
What am I doing wrong?
In the newest GIMP the default Crop Tool behaviour has changed to allow non-destructive cropping. Basically it crops the canvas, not the layers, and so doesn't delete any pixels. If you export the image however, the crop should be applied to the exported file. I just tested a non-destructive crop and exported the result as PNG and it works as expected. I can't reproduce the problem you describe.
If you want to crop the image, and delete the pixels like in older versions of GIMP, choose the "Delete cropped pixels" option in the Tool Options
