convert command that you used didn’t actually convert your image down to a 16-bit colour RGB image. It just reduced the number of colours in the palette, which can be done with the Gimp!
To reduce the colour palette you have to go to “Image -> Mode -> Indexed”, select “Generate optimum palette”, and set the Maximum number of colours to 256.
I found that the “Floyd-Steinberg (reduced colour bleeding)” ditherer worked the best on your image.
Once you save it as a bitmap,
identify -verbose $filename will give you the following results on the new image.
You’ll see that the original, the convert-reduced-palette, and the gimp-reduced-palette images are all 24-bit, although the one done with the Gimp will say that it’s Type is “Palette”, rather than TrueColor.
Doesn’t really matter too much though - as long as the image looks fine!