I followed this tutorial on GIMP to make an image transparent. Here is a snippet of the original image opened inside Windows Image Viewer (so the grey background is part of Windows Image Viewer, I wanted the white background to show up which was hard to notice if I had just uploaded the original image here rather than a snippet image),
Inside GIMP the image appears to be transparent (again this is a snippet so the grey background is not part of the image),
But when I export the image and open it again in Windows Image Viewer the white background is still present. I need the white background to disappear so that when I add it as an icon in a program it blends in seamlessly.
Kudos to Volker Siegel!
It turns out I was exporting as a JPG. After exporting as a PNG I got the full transparency shown here in MS Image Viewer again (so MS IV has a greyish background and you can see the image is transparent),
You do not write which way you export the image, but this is important.
It needs to be exported to an image format that supports saving the transparency information to the file.
Otherwise, the export process has to find a way to save the image without the transparency - asking you, or just pretending the background is white, for example.
The most common image file format supporting transparency is PNG,
and the most common format not supporting it is JPG.
The problem could be that the image was exported to JPG, and lost the transparency while exporting.
If that's right, exporting to PNG instead should help.
As a side note, it's interesting that the handling of transparency was not "just left out" from JPEG - it's more like the whole concept of lossy compressing does not go together with the idea of transparency.
The basic idea of lossy compression is to store information that is more visible in a more detailed way, and leave out information that is less visible. But if a pixel can be transparent, we do not know about what is visible - it depends on the background that we do not have.