OK, you got me convinced. It remains the spots are the image of pixels that all the very same specific color, and this color happens to be the first in the color map, and is reported as the “Background color” by identify -verbose.
So what can I do?
For asmall image like that, you can use the highest quality JPEG with no chroma sub-sampling, i.e. 4:4:4 (1x1).
FastStone Viewer can even store a JPEG as RGB with no compression at all!
Which then leaves the question of why that converter site would make such a silly choice… 
Hi Steve,
is this the only GIF you want to make?
It’s really easy to do this in GIMP and I am tempted to make a litle video about it if you have to do this more than once.
@rstein Yes but I will have to make others.
Learn to use Gimp then. Much better than what you did.