I tried to convert the black and white colors of the first picture into the second one but I get artifacts as you can see around the edges. How can I remove those artifacts using Gimp?
Open the original image in GIMP, and select the transparent pixel lock in the layers panel. You need to do this or you will mess up the antialiasing, and the edges will look jagged.
Select the Bucket Fill tool, and in the tool options, set the threshold slider about half way. Then fill the image with the colours you want. Note the image is in Greyscale mode, so if you want to add colours, you'll need to convert to RGB first (Image > Mode > RGB).
In this screenshot I have circled the transparent pixel lock and Bucket Fill tool threshold settings in red.
A slightly left-field answer to your question, but I would simply redraw the image as a vector. It's only three circles.
It would take less than a minute, and be absolutely spot on. You could easily adjust colours etc going forwards, and use it as an SVG icon, and a variety of other vector-tastic bonuses :)